Std10-Determination of Residual Solvent残留溶剂(中国药典CP 英文版)
467 残留溶剂 usp
<467> RESIDUAL SOLVENTS残留溶剂(Chapter under this new title—to become official July 1, 2008)（这个新名称下的通则----2008年7月1日起生效）(Current chapter title is <467> Organic Volatile Impurities)（当前通则名称是<467>有机挥发性杂质）INTRODUCTION介绍This general chapter applies to existing drug substances, excipients, and products. All substances and products are subject to relevant control of solvents likely to be present in a substance or product.本通则适用于现有原料药、辅助剂、成药。
所有原料药和成药均需对可能存在其中的溶剂进行控制。
Where the limits to be applied comply with those given below, tests for residual solvents are not generally mentioned in specific monographs because the solvents employed may vary from one manufacturer to another.在所适用的限度与下面所述相符的情况下，残留溶剂检测一般不在具体各论中提及，因为每个生产商所使用的溶剂可能都不相同。
The objective of this general chapter is to provide acceptable amounts of residual solvents in pharmaceuticals for the safety of the patient. The general chapter recommends the use of less toxic solvents and describes levels considered to be toxicologically acceptable for some residual solvents.本通则的目的是为了保障病人的安全，提供在药物中残留溶剂的可接受数量。
格拉布斯法—异常值判断
格拉布斯法—判断(2009-04-0716:38:20)标签：▲概述：一组测量数据中，如果个别数据偏离平均值很远，那么这个(这些)数据称作“可疑值”。
如果用统计方法—例如格拉布斯(Grubbs)法判断，能将“可疑值”从此组测量数据中剔除而参与平均值的计算，那么该“可疑值”就称作“(粗大误差)”。
本文就是介绍如何用格拉布斯法判断“可疑值”是否为“”。
▲测量数据：例如测量10次(n＝10)，获得以下数据：、、、、、、、、、。
▲排列数据：将上述测量数据按从小到大的顺序排列，得到、、、、、、、、、。
可以肯定，可疑值是最小值就是最大值。
▲计算平均值x-和标准差s：x-＝；标准差s＝。
计算时，必须将所有10个数据全部包含在内。
▲计算偏离值：平均值与最小值之差为－＝；最大值与平均值之差为－＝。
▲确定一个可疑值：比较起来，最大值与平均值之差大于平均值与最小值之差，因此认为最大值是可疑值。
▲计算G i值：G i＝(x i－x-)/s；其中i是可疑值的排列序号＝(x10－x-)/s＝－/＝。
由于x10－x-是残差，而s是标准差，——10号；因此G10因而可认为G是残差与标准差的比值。
下面要把计算值G i与格拉布斯表给出的10临界值G P(n)比较，如果计算的G i值大于表中的临界值G P(n)，则能判断该测量数据是，可以剔除。
但是要提醒，临界值G P(n)与两个参数有关：检出水平α(与置信概率P有关)和测量次数n(与自由度f有关)。
▲定检出水平α：如果要求严格，检出水平α可以定得小一些，例如定α＝，那么置信概率P＝1－α＝；如果要求严格，α可以定得大一些，例如定α＝，即P＝；通常定α＝，P＝。
▲查格拉布斯表获得临界值：根据选定的P值(此处为和测量次数n(此处为10)，查格拉布斯表，横竖相交得临界值G95(10)＝。
▲比较计算值G i和临界值G95(10)：G i＝，G95(10)＝，G i＞G95(10)。
▲判断是否为：因为G i＞G95(10)，可以判断测量值为，将它从10个测量数据中剔除。
ASTM D256-2010
Designation:D256–10Standard Test Methods forDetermining the Izod Pendulum Impact Resistance of Plastics1This standard is issued under thefixed designation D256;the number immediately following the designation indicates the year of original adoption or,in the case of revision,the year of last revision.A number in parentheses indicates the year of last reapproval.A superscript epsilon(´)indicates an editorial change since the last revision or reapproval.This standard has been approved for use by agencies of the Department of Defense.1.Scope*1.1These test methods cover the determination of the resistance of plastics to“standardized”(see Note1)pendulum-type hammers,mounted in“standardized”machines,in break-ing standard specimens with one pendulum swing(see Note2). The standard tests for these test methods require specimens made with a milled notch(see Note3).In Test Methods A,C, and D,the notch produces a stress concentration that increases the probability of a brittle,rather than a ductile,fracture.In Test Method E,the impact resistance is obtained by reversing the notched specimen180°in the clamping vise.The results of all test methods are reported in terms of energy absorbed per unit of specimen width or per unit of cross-sectional area under the notch.(See Note4.)N OTE1—The machines with their pendulum-type hammers have been “standardized”in that they must comply with certain requirements, including afixed height of hammer fall that results in a substantiallyfixed velocity of the hammer at the moment of impact.However,hammers of different initial energies(produced by varying their effective weights)are recommended for use with specimens of different impact resistance. Moreover,manufacturers of the equipment are permitted to use different lengths and constructions of pendulums with possible differences in pendulum rigidities resulting.(See Section5.)Be aware that other differences in machine design may exist.The specimens are“standard-ized”in that they are required to have onefixed length,onefixed depth, and one particular design of milled notch.The width of the specimens is permitted to vary between limits.N OTE2—Results generated using pendulums that utilize a load cell to record the impact force and thus impact energy,may not be equivalent to results that are generated using manually or digitally encoded testers that measure the energy remaining in the pendulum after impact.N OTE3—The notch in the Izod specimen serves to concentrate the stress,minimize plastic deformation,and direct the fracture to the part of the specimen behind the notch.Scatter in energy-to-break is thus reduced. However,because of differences in the elastic and viscoelastic properties of plastics,response to a given notch varies among materials.A measure of a plastic’s“notch sensitivity”may be obtained with Test Method D by comparing the energies to break specimens having different radii at the base of the notch.N OTE4—Caution must be exercised in interpreting the results of these standard test methods.The following testing parameters may affect test results significantly:Method of fabrication,including but not limited to processingtechnology,molding conditions,mold design,and thermaltreatments;Method of notching;Speed of notching tool;Design of notching apparatus;Quality of the notch;Time between notching and test;Test specimen thickness,Test specimen width under notch,andEnvironmental conditioning.1.2The values stated in SI units are to be regarded as standard.The values given in parentheses are for information only.1.3This standard does not purport to address all of the safety concerns,if any,associated with its use.It is the responsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.N OTE5—These test methods resemble ISO180:1993in regard to title only.The contents are significantly different.2.Referenced Documents2.1ASTM Standards:2D618Practice for Conditioning Plastics for TestingD883Terminology Relating to PlasticsD3641Practice for Injection Molding Test Specimens of Thermoplastic Molding and Extrusion MaterialsD4066Classification System for Nylon Injection and Ex-trusion Materials(PA)D5947Test Methods for Physical Dimensions of Solid Plastics Specimens1These test methods are under the jurisdiction of ASTM Committee D20on Plastics and are the direct responsibility of Subcommittee D20.10on MechanicalProperties.Current edition approved May1,2010.Published June2010.Originally approved st previous edition approved in2006as D256-06a´1.DOI: 10.1520/D0256-10.2For referenced ASTM standards,visit the ASTM website,,or contact ASTM Customer Service at service@.For Annual Book of ASTM Standards volume information,refer to the standard’s Document Summary page on the ASTM website.*A Summary of Changes section appears at the end of this standard. Copyright©ASTM International,100Barr Harbor Drive,PO Box C700,West Conshohocken,PA19428-2959,United States.D6110Test Method for Determining the Charpy Impact Resistance of Notched Specimens of PlasticsE691Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method2.2ISO Standard:ISO180:1993Plastics—Determination of Izod Impact Strength of Rigid Materials33.Terminology3.1Definitions—For definitions related to plastics see Terminology D883.3.2Definitions of Terms Specific to This Standard:3.2.1cantilever—a projecting beam clamped at only one end.3.2.2notch sensitivity—a measure of the variation of impact energy as a function of notch radius.4.Types of Tests4.1Four similar methods are presented in these test meth-ods.(See Note6.)All test methods use the same testing machine and specimen dimensions.There is no known means for correlating the results from the different test methods.N OTE6—Previous versions of this test method contained Test Method B for Charpy.It has been removed from this test method and has been published as D6110.4.1.1In Test Method A,the specimen is held as a vertical cantilever beam and is broken by a single swing of the pendulum.The line of initial contact is at afixed distance from the specimen clamp and from the centerline of the notch and on the same face as the notch.4.1.2Test Method C is similar to Test Method A,except for the addition of a procedure for determining the energy ex-pended in tossing a portion of the specimen.The value reported is called the“estimated net Izod impact resistance.”Test Method C is preferred over Test Method A for materials that have an Izod impact resistance of less than27J/m(0.5 ft·lbf/in.)under notch.(See Appendix X4for optional units.) The differences between Test Methods A and C become unimportant for materials that have an Izod impact resistance higher than this value.4.1.3Test Method D provides a measure of the notch sensitivity of a material.The stress-concentration at the notch increases with decreasing notch radius.4.1.3.1For a given system,greater stress concentration results in higher localized rates-of-strain.Since the effect of strain-rate on energy-to-break varies among materials,a mea-sure of this effect may be obtained by testing specimens with different notch radii.In the Izod-type test it has been demon-strated that the function,energy-to-break versus notch radius, is reasonably linear from a radius of0.03to2.5mm(0.001to 0.100in.),provided that all specimens have the same type of break.(See5.8and22.1.)4.1.3.2For the purpose of this test,the slope,b(see22.1), of the line between radii of0.25and1.0mm(0.010and0.040 in.)is used,unless tests with the1.0-mm radius give“non-break”results.In that case,0.25and0.50-mm(0.010and 0.020-in.)radii may be used.The effect of notch radius on the impact energy to break a specimen under the conditions of this test is measured by the value b.Materials with low values of b, whether high or low energy-to-break with the standard notch, are relatively insensitive to differences in notch radius;while the energy-to-break materials with high values of b is highly dependent on notch radius.The parameter b cannot be used in design calculations but may serve as a guide to the designer and in selection of materials.4.2Test Method E is similar to Test Method A,except that the specimen is reversed in the vise of the machine180°to the usual striking position,such that the striker of the apparatus impacts the specimen on the face opposite the notch.(See Fig. 1,Fig.2.)Test Method E is used to give an indication of the unnotched impact resistance of plastics;however,results ob-tained by the reversed notch method may not always agree with those obtained on a completely unnotched specimen.(See 28.1.)4,55.Significance and Use5.1Before proceeding with these test methods,reference should be made to the specification of the material being tested. Any test specimen preparation,conditioning,dimensions,and testing parameters covered in the materials specification shall take precedence over those mentioned in these test methods.If there is no material specification,then the default conditions apply.5.2The pendulum impact test indicates the energy to break standard test specimens of specified size under stipulated parameters of specimen mounting,notching,and pendulum velocity-at-impact.3Available from American National Standards Institute(ANSI),25W.43rd St., 4th Floor,New York,NY10036,.4Supporting data giving results of the interlaboratory tests are available from ASTM Headquarters.Request RR:D20-1021.5Supporting data giving results of the interlaboratory tests are available from ASTM Headquarters.RequestRR:D20-1026.FIG.1Relationship of Vise,Specimen,and Striking Edge to Each Other for Izod Test Methods A andC5.3The energy lost by the pendulum during the breakage of the specimen is the sum of the following:5.3.1Energy to initiate fracture of the specimen;5.3.2Energy to propagate the fracture across the specimen;5.3.3Energy to throw the free end (or ends)of the broken specimen (“toss correction”);5.3.4Energy to bend the specimen;5.3.5Energy to produce vibration in the pendulum arm;5.3.6Energy to produce vibration or horizontal movement of the machine frame or base;5.3.7Energy to overcome friction in the pendulum bearing and in the indicating mechanism,and to overcome windage (pendulum air drag);5.3.8Energy to indent or deform plastically the specimen at the line of impact;and5.3.9Energy to overcome the friction caused by the rubbing of the striker (or other part of the pendulum)over the face of the bent specimen.5.4For relatively brittle materials,for which fracture propa-gation energy is small in comparison with the fracture initiation energy,the indicated impact energy absorbed is,for all practical purposes,the sum of factors 5.3.1and 5.3.3.The toss correction (see 5.3.3)may represent a very large fraction of the total energy absorbed when testing relatively dense and brittle materials.Test Method C shall be used for materials that have an Izod impact resistance of less than 27J/m (0.5ft·lbf/in.).(See Appendix X4for optional units.)The toss correction obtained in Test Method C is only an approximation of the toss error,since the rotational and rectilinear velocities may not be the same during the re-toss of the specimen as for the original toss,and because stored stresses in the specimen may have been released as kinetic energy during the specimen fracture.5.5For tough,ductile,fiber filled,or cloth-laminated mate-rials,the fracture propagation energy (see 5.3.2)may be large compared to the fracture initiation energy (see 5.3.1).When testing these materials,factors (see 5.3.2,5.3.5,and 5.3.9)canbecome quite significant,even when the specimen is accurately machined and positioned and the machine is in good condition with adequate capacity.(See Note 7.)Bending (see 5.3.4)and indentation losses (see 5.3.8)may be appreciable when testing soft materials.N OTE 7—Although the frame and base of the machine should be sufficiently rigid and massive to handle the energies of tough specimens without motion or excessive vibration,the design must ensure that the center of percussion be at the center of strike.Locating the striker precisely at the center of percussion reduces vibration of the pendulum arm when used with brittle specimens.However,some losses due to pendulum arm vibration,the amount varying with the design of the pendulum,will occur with tough specimens,even when the striker is properly positioned.5.6In a well-designed machine of sufficient rigidity and mass,the losses due to factors 5.3.6and 5.3.7should be very small.Vibrational losses (see 5.3.6)can be quite large when wide specimens of tough materials are tested in machines of insufficient mass,not securely fastened to a heavy base.5.7With some materials,a critical width of specimen may be found below which specimens will appear ductile,as evidenced by considerable drawing or necking down in the region behind the notch and by a relatively high-energy absorption,and above which they will appear brittle as evidenced by little or no drawing down or necking and by a relatively low-energy absorption.Since these methods permit a variation in the width of the specimens,and since the width dictates,for many materials,whether a brittle,low-energy break or a ductile,high energy break will occur,it is necessary that the width be stated in the specification covering that material and that the width be reported along with the impact resistance.In view of the preceding,one should not make comparisons between data from specimens having widths that differ by more than a few mils.5.8The type of failure for each specimen shall be recorded as one of the four categories listed as follows:C =Complete Break —A break where the specimen separates into two or more pieces.H =Hinge Break —An incomplete break,such that one part of the specimen cannot support itself above the horizontal when the other part is held vertically (less than 90°included angle).P =Partial Break —An incomplete break that does not meet the definition for a hinge break but has frac-tured at least 90%of the distance between the vertex of the notch and the opposite side.NB =Non-Break —An incomplete break where the frac-ture extends less than 90%of the distance be-tween the vertex of the notch and the opposite side.For tough materials,the pendulum may not have the energy necessary to complete the breaking of the extreme fibers and toss the broken piece or pieces.Results obtained from “non-break”specimens shall be considered a departure from stan-dard and shall not be reported as a standard result.Impact resistance cannot be directly compared for any two materials that experience different types of failure as defined in the test method by this code.Averages reported must likewise be derived from specimens contained within a single failure category.This letter code shall suffix the reported impact identifying the types of failure associated with the reported value.If more than one type of failure is observed for asampleFIG.2Relationship of Vise,Specimen,and Striking Edge to EachOther for Test MethodEmaterial,then the report will indicate the average impact resistance for each type of failure,followed by the percent of the specimens failing in that manner and suffixed by the letter code.5.9The value of the impact methods lies mainly in the areas of quality control and materials specification.If two groups of specimens of supposedly the same material show significantly different energy absorptions,types of breaks,critical widths,or critical temperatures,it may be assumed that they were made of different materials or were exposed to different processing or conditioning environments.The fact that a material shows twice the energy absorption of another under these conditions of test does not indicate that this same relationship will exist under another set of test conditions.The order of toughness may even be reversed under different testing conditions.N OTE8—A documented discrepancy exists between manual and digital impact testers,primarily with thermoset materials,including phenolics, having an impact value of less than54J/m(1ft-lb/in.).Comparing data on the same material,tested on both manual and digital impact testers, may show the data from the digital tester to be significantly lower than data from a manual tester.In such cases a correlation study may be necessary to properly define the true relationship between the instruments.TEST METHOD A—CANTILEVER BEAM TEST6.Apparatus6.1The machine shall consist of a massive base on which is mounted a vise for holding the specimen and to which is connected,through a rigid frame and bearings,a pendulum-type hammer.(See 6.2.)The machine must also have a pendulum holding and releasing mechanism and a mechanism for indicating the breaking energy of the specimen.6.2A jig for positioning the specimen in the vise and graphs or tables to aid in the calculation of the correction for friction and windage also should be included.One type of machine is shown in Fig.3.One design of specimen-positioning jig is illustrated in Fig.4.Detailed requirements are given in subsequent paragraphs.General test methods for checking and calibrating the machine are given in Appendix X2.Additional instructions for adjusting a particular machine should be supplied by the manufacturer.6.3The pendulum shall consist of a single or multi-membered arm with a bearing on one end and a head, containing the striker,on the other.The arm must be suffi-ciently rigid to maintain the proper clearances and geometric relationships between the machine parts and the specimen and to minimize vibrational energy losses that are always includedin the measured impact resistance.Both simple and compound pendulum designs may comply with this test method.6.4The striker of the pendulum shall be hardened steel and shall be a cylindrical surface having a radius of curvature of 0.8060.20mm(0.03160.008in.)with its axis horizontal and perpendicular to the plane of swing of the pendulum.The line of contact of the striker shall be located at the center of percussion of the pendulum within62.54mm(60.100in.) (See Note9.)Those portions of the pendulum adjacent to the cylindrical striking edge shall be recessed or inclined at a suitable angle so that there will be no chance for other than this cylindrical surface coming in contact with the specimen during the break.N OTE9—The distance from the axis of support to the center of percussion may be determined experimentally from the period of small amplitude oscillations of the pendulum by means of the following equation:L5~g/4p2!p2FIG.3Cantilever Beam(Izod-Type)ImpactMachine FIG.4Jig for Positioning Specimen forClampingwhere:L=distance from the axis of support to the center of percussion,m or(ft),g=local gravitational acceleration(known to an accuracy of one part in one thousand),m/s2or(ft/s2),p= 3.1416(4p2=39.48),andp=period,s,of a single complete swing(to and fro)determined by averaging at least20consecutive and uninterrupted swings.Theangle of swing shall be less than5°each side of center.6.5The position of the pendulum holding and releasing mechanism shall be such that the vertical height of fall of the striker shall be61062mm(24.060.1in.).This will produce a velocity of the striker at the moment of impact of approxi-mately3.5m(11.4ft)/s.(See Note10.)The mechanism shall be so constructed and operated that it will release the pendulum without imparting acceleration or vibration to it.N OTE10—V5~2gh!0.5where:V=velocity of the striker at the moment of impact(m/s),g=local gravitational acceleration(m/s2),andh=vertical height of fall of the striker(m).This assumes no windage or friction.6.6The effective length of the pendulum shall be between 0.33and0.40m(12.8and16.0in.)so that the required elevation of the striker may be obtained by raising the pendulum to an angle between60and30°above the horizontal.6.7The machine shall be provided with a basic pendulum capable of delivering an energy of2.760.14J(2.0060.10 ft·lbf).This pendulum shall be used with all specimens that extract less than85%of this energy.Heavier pendulums shall be provided for specimens that require more energy to break. These may be separate interchangeable pendulums or one basic pendulum to which extra pairs of equal calibrated weights may be rigidly attached to opposite sides of the pendulum.It is imperative that the extra weights shall not significantly change the position of the center of percussion or the free-hanging rest point of the pendulum(that would consequently take the machine outside of the allowable calibration tolerances).A range of pendulums having energies from2.7to21.7J(2to16 ft·lbf)has been found to be sufficient for use with most plastic specimens and may be used with most machines.A series of pendulums such that each has twice the energy of the next will be found convenient.Each pendulum shall have an energy within60.5%of its nominal capacity.6.8A vise shall be provided for clamping the specimen rigidly in position so that the long axis of the specimen is vertical and at right angles to the top plane of the vise.(See Fig.1.)This top plane shall bisect the angle of the notch with a tolerance of0.12mm(0.005in.).Correct positioning of the specimen is generally done with a jig furnished with the machine.The top edges of thefixed and moveable jaws shall have a radius of0.2560.12mm(0.01060.005in.).For specimens whose thickness approaches the lower limiting value of3.00mm(0.118in.),means shall be provided to prevent the lower half of the specimen from moving during the clamping or testing operations(see Fig.4and Note11.)N OTE11—Some plastics are sensitive to clamping pressure;therefore,cooperating laboratories should agree upon some means of standardizing the clamping force.One method is using a torque wrench on the screw of the specimen vise.If the faces of the vise or specimen are notflat and parallel,a greater sensitivity to clamping pressure may be evident.See the calibration procedure in Appendix X2for adjustment and correction instructions for faulty instruments.6.9When the pendulum is free hanging,the striking surface shall come within0.2%of scale of touching the front face of a standard specimen.During an actual swing this element shall make initial contact with the specimen on a line22.0060.05 mm(0.8760.002in.)above the top surface of the vise. 6.10Means shall be provided for determining the energy expended by the pendulum in breaking the specimen.This is accomplished using either a pointer and dial mechanism or an electronic system consisting of a digital indicator and sensor (typically an encoder or resolver).In either case,the indicated breaking energy is determined by detecting the height of rise of the pendulum beyond the point of impact in terms of energy removed from that specific pendulum.Since the indicated energy must be corrected for pendulum-bearing friction, pointer friction,pointer inertia,and pendulum windage,in-structions for making these corrections are included in10.3and Annex A1and Annex A2.If the electronic display does not automatically correct for windage and friction,it shall be incumbent for the operator to determine the energy loss manually.(See Note12.)N OTE12—Many digital indicating systems automatically correct for windage and friction.The equipment manufacturer may be consulted for details concerning how this is performed,or if it is necessary to determine the means for manually calculating the energy loss due to windage and friction.6.11The vise,pendulum,and frame shall be sufficiently rigid to maintain correct alignment of the hammer and speci-men,both at the moment of impact and during the propagation of the fracture,and to minimize energy losses due to vibration. The base shall be sufficiently massive that the impact will not cause it to move.The machine shall be so designed,con-structed,and maintained that energy losses due to pendulum air drag(windage),friction in the pendulum bearings,and friction and inertia in the indicating mechanism are held to a minimum.6.12A check of the calibration of an impact machine is difficult to make under dynamic conditions.The basic param-eters are normally checked under static conditions;if the machine passes the static tests,then it is assumed to be accurate.The calibration procedure in Appendix X2should be used to establish the accuracy of the equipment.However,for some machine designs it might be necessary to change the recommended method of obtaining the required calibration measurements.Other methods of performing the required checks may be substituted,provided that they can be shown to result in an equivalent accuracy.Appendix X1also describes a dynamic test for checking certain features of the machine and specimen.6.13Micrometers—Apparatus for measurement of the width of the specimen shall comply with the requirements of Test Methods D5947.Apparatus for the measurement of the depth of plastic material remaining in the specimen under the notch shall comply with requirements of Test Methods D5947, provided however that the one anvil or presser foot shall beatapered blade conforming to the dimensions given in Fig.5.The opposing anvil or presser foot shall be flat and conforming to Test Methods D5947.7.Test Specimens7.1The test specimens shall conform to the dimensions and geometry of Fig.6,except as modified in accordance with 7.2,7.3,7.4,and 7.5.To ensure the correct contour and conditions of the specified notch,all specimens shall be notched as directed in Section 8.7.1.1Studies have shown that,for some materials,the location of the notch on the specimen and the length of the impacted end may have a slight effect on the measured impact resistance.Therefore,unless otherwise specified,care must be taken to ensure that the specimen conforms to the dimensions shown in Fig.6and that it is positioned as shown in Fig.1or Fig.2.7.2Molded specimens shall have a width between 3.0and 12.7mm (0.118and 0.500in.).Use the specimen width as specified in the material specification or as agreeduponN OTE 1—These views not to scale.N OTE 2—Micrometer to be satin-chrome finished with friction thimble.N OTE 3—Special anvil for micrometer caliper 0to 25.4mm range (50.8mm frame)(0to 1in.range (2-in.frame)).N OTE 4—Anvil to be oriented with respect to frame as shown.N OTE 5—Anvil and spindle to have hardened surfaces.N OTE 6—Range:0to 25.4mm (0to 1in.in thousandths of an inch).N OTE 7—Adjustment must be at zero when spindle and anvil are in contact.FIG.5Early (ca.1970)Version of a Notch-DepthMicrometerbetween the supplier and the customer.All specimens having one dimension less than 12.7mm (0.500in.)shall have the notch cut on the shorter side.Otherwise,all compression-molded specimens shall be notched on the side parallel to the direction of application of molding pressure.(See Fig.6.)N OTE 13—While subsection 7.5requires perpendicular pairs of plane parallel surfaces,the common practice has been to accept the non-parallel drafted surfaces formed when directly injection molding specimens for Izod ers must be aware that employing a trapezoidal section rather than a rectangular section may lead to data shifts and scatter.Unequal stress,created by clamping in the fracture region and dynamic twisting,caused by uneven striking of the specimen are prone to occur when the faces of the specimen are not parallel.Interlaboratory compari-sons must clearly spell out the specimen preparation conditions.7.2.1Extreme care must be used in handling specimens less than 6.35mm (0.250in.)wide.Such specimens must be accurately positioned and supported to prevent twist or lateral buckling during the test.Some materials,furthermore,are very sensitive to clamping pressure (see Note 11).7.2.2A critical investigation of the mechanics of impact testing has shown that tests made upon specimens under 6.35mm (0.250in.)wide absorb more energy due to crushing,bending,and twisting than do wider specimens.Therefore,specimens 6.35mm (0.250in.)or over in width are recom-mended.The responsibility for determining the minimumspecimen width shall be the investigator’s,with due reference to the specification for that material.7.2.3Material specification should be consulted for pre-ferred molding conditions.The type of mold and molding machine used and the flow behavior in the mold cavity will influence the impact resistance obtained.A specimen taken from one end of a molded plaque may give different results than a specimen taken from the other end.Cooperating laboratories should therefore agree on standard molds con-forming to the material specification.Practice D3641can be used as a guide for general molding tolerances,but refer to the material specification for specific molding conditions.7.2.4The impact resistance of a plastic material may be different if the notch is perpendicular to,rather than parallel to,the direction of molding.The same is true for specimens cut with or across the grain of an anisotropic sheet or plate.7.3For sheet materials,the specimens shall be cut from the sheet in both the lengthwise and crosswise directions unless otherwise specified.The width of the specimen shall be the thickness of the sheet if the sheet thickness is between 3.0and 12.7mm (0.118and 0.500in.).Sheet material thicker than 12.7mm shall be machined down to 12.7mm.Specimens with a 12.7-mm square cross section may be tested either edgewise or flatwise as cut from the sheet.When specimens are tested flatwise,the notch shall be made on the machined surface ifthemmin.A 10.1660.050.40060.002B 31.861.0 1.2560.04C 63.562.0 2.5060.08D 0.25R 60.050.010R 60.002E12.7060.200.50060.008FIG.6Dimensions of Izod-Type TestSpecimen。
关于有效自由度大于10的探讨
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1 有效自由度 eff的计算公式
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一类离散神经系统中的混沌存在性
第50卷第2期2023年北京化工大学学报(自然科学版)Journal of Beijing University of Chemical Technology (Natural Science)Vol.50,No.22023引用格式:陈晓楠,常玉,陈娇,等.一类离散神经系统中的混沌存在性[J].北京化工大学学报(自然科学版),2023,50(2):119-125.CHEN XiaoNan,CHANG Yu,CHEN Jiao,et al.Chaos in a discrete FitzHugh -Nagumo model[J].Journal of Beijing U⁃niversity of Chemical Technology (Natural Science),2023,50(2):119-125.一类离散神经系统中的混沌存在性陈晓楠 常 玉* 陈 娇 杨睿丰(北京化工大学数理学院,北京 100029)摘 要:研究了离散FitzHugh -Nagumo 神经系统的混沌现象㊂从理论上严格证明了在一定参数条件下系统存在Marotto 意义下的混沌,并通过数值计算模拟出该混沌吸引子,以及周期14轨㊁周期16轨等其他复杂动态㊂此外,通过Lyapunov 指数的计算对Marotto 混沌吸引子的存在性加以验证㊂关键词:离散FitzHugh -Nagumo 系统;Marotto 混沌;Lyapunov 指数中图分类号:O193 DOI :10.13543/j.bhxbzr.2023.02.015收稿日期:2022-01-05基金项目:国家自然科学基金(11771033)第一作者:女,1996年生,硕士生*通信联系人E⁃mail:changyu@引 言神经系统主导着生物体生理活动的调节,其功能主要通过神经元产生㊁处理和传递电信号来实现[1-2]㊂1952年,基于乌贼巨型轴突上电位变化的实验数据,Hodgkin 等[3]建立了一个四维非线性连续数学模型 Hodgkin -Huxley 模型(HH 模型),开启了对神经元电活动的数学研究㊂随后,FitzHugh [4]在保留系统主要动态的基础上,将HH 模型简化为二维系统;之后Nagumo 等[5]通过电路实验成功实现了该模型的动态,因此该模型也被称为FitzHugh -Nagumo 模型(FHN 模型)㊂由于系统的复杂性,针对FHN 模型的动态研究一直备受关注㊂Hayashi [6]给出了FHN 模型非平凡闭轨唯一性的充分条件㊂Rocşoreanu 等[7]证明了FHN 模型中saddle⁃node 分支㊁Hopf 分支以及Bogdanov -Takens分支的存在性㊂Jing 等[8]对FHN 模型应用Euler 方法得到离散FHN 系统,证明了该系统存在周期10轨㊁倍周期级联以及混沌等复杂动态㊂之后,Zhan 等[9]推广了一类离散FHN 模型,通过数值模拟分析了系统产生混沌的内外机制,以及延滞反馈项㊁高斯白色噪声项和磁场对激发神经元放电模式的影响㊂随着对该模型研究的深入,大量结果表明模型的动态对揭示现实中神经元放电活动的机制具有重要的参考价值㊂本文以文献[9]中的离散FHN 模型为研究对象,从理论上分析了系统中Marotto 混沌的存在性,得到了这类混沌存在的参数充分条件,并通过数值模拟展示了混沌吸引子㊁高周期解等复杂的非线性动态,这些结果丰富了对该模型的研究㊂1 离散FHN 模型1.1 模型描述考虑如下离散FHN 模型[9]f :æèçöø÷x y →x +(δx -x 33-y +)I y +δτ(ax +b -y æèççççöø÷÷÷÷)(1)式中,x 表示神经元膜电位;y 表示与膜电位恢复到静息电位过程有关的恢复变量;参数I (∈R )表示神经元接受的外界刺激强度或输入电流,其他系统参数a (>0)㊁b (∈R )㊁τ(>0)㊁δ(0<δ<1)的说明参见文献[9]㊂1.2 Marotto 意义下混沌的存在性在高维离散系统中存在的最典型的混沌吸引子是Marotto 混沌,这类混沌系统的动态十分复杂,如:该系统存在任意正整数周期的周期轨;同时拥有 scrambled”集,这是一个不包括周期点㊁不可数的㊁具有初值敏感性㊁没有渐近周期点的不变集㊂此类混沌的存在性完全由系统是否拥有某类特殊不动点即snap⁃back repeller 所决定[10-11]㊂本文将从理论上研究Marotto 意义下的混沌在系统(1)中的参数存在条件㊂应用离散动力系统的定性理论可得,当b =I ,0<a <1时,系统(1)存在一个不动点Z 0(x 0,y 0)=(3(1-a ),a3(1-a )+b )㊂此不动点Z 0是snap⁃back repeller 需要满足以下条件:①存在Z 0的一个邻域U Z 0,在此邻域中Z 0是expanding 不动点,即∀Z (x ,y )∈U Z 0,Z 所有特征值的模长均大于1;②U Z 0中存在一点Z *(x *,y *)≠Z 0,满足f M (Z *)=Z 0,|D f M (Z *)|≠0,M ∈N +㊂下面讨论Z 0(x 0,y 0)是snap⁃back repeller,即满足条件①㊁②的参数充分条件㊂引理1摇若23<a <1(2)2a -13a -2<τ<12-a -22a (1-a )(3)τ(2-3a )+12(1-a )<δ<1(4)则(ⅰ)存在点集U {=(x ,y )|x (∈1+1-2aττ,1+1-δaδ-)τ,y ∈}R ,该集合中任意点的特征值都是一对共轭复数,且模长大于1;(ⅱ)存在不动点Z 0(x 0,y 0)=(3(1-a ),a3(1-a )+b )的邻域U Z 0{=(x ,y )|(x -x 0)2r 2x 0+(y -y 0)2r 2y 0≤1,r x 0,r y 0∈R }+,满足UZ 0⊂U ㊂证明:(ⅰ)首先证明2a -13a -2<12-a -22a (1-a )㊂因为23<a <1,所以2a -13a -2=3a -2+1-a 3a -2=1+1-a3a -2>1㊂由(a (1-a )-(2a -1)2a (1-a ))㊃(a (1-a )+(2a -1)2a (1-a ))=(a (1-a ))2-((2a -1)2a (1-a ))2=-9a (1-a )(a -23()a -)13<0,可得a (1-a )-(2a -1)2a (1-a )<0㊂又因为(2-a -22a (1-a ))(2-a +22a (1-a ))=(2-a )2-(22a (1-a ))2=(9a -)232>0,故2-a -22a (1-a )>0,从而有2a -13a -2-12-a -22a (1-a )=2(a (1-a )-(2a -1)2a (1-a ))(3a -2)(2-a -22a (1-a ))<0㊂其次证明1+1-δa δ-τ>1+1-2aττ>0㊂显然τ+1>2τ>2aτ,故1+1-2aττ>0㊂由式(2)以及τ>2a -13a -2,可得τ-11-a >2a -13a -2-11-a=13a -2>1㊂又由2a -13a -2>1>δ,可知τ-11-a >δ,τ>δ,从而有1+1-δaδ-τ=δ(1-a )+1-τδ-τ=(1-a ()τ-11-a-)δτ-δ>0㊂因为2a -13a -2-1a =2(a -1)2a (3a -2)>0,所以2a -13a -2>1a ㊂由τ>2a -13a -2>1a ,可得aτ>1,从而2τ1-aτ<0㊂(注意到1+1-2aτ)τ(-1+1-δa δ-)τ=(1-aτ)(2δ-2τ1-a )ττ(δ-τ)<0,故1+1-2aττ<1+1-δaδ-τ㊂又因为1-δa δ-τ<0<1+2aττ,所以1+1-δaδ-τ<1+1+2aττ㊂综上可得,点集U {=(x ,y )|x ∈㊃021㊃北京化工大学学报(自然科学版) 2023年(1+1-2aττ,1+1-δaδ-)τ,y ∈}R 是存在的㊂任取Z (x ,y )∈U ,下面证明Z (x ,y )具有一对共轭复特征值,且模长大于1㊂易知系统(1)在点Z (x ,y )处的Jacobian 矩阵为J =1+δ(1-x 2)-δδa τ1-δæèççöø÷÷τ其特征方程为λ2+p (x )λ+q (x )=0,其中p (x )=-2+δτ-δ(1-x 2),q (x )=1-δτ+δ(1-x 2)-δ2τ(1-x 2)+δ2a τ,特征值为λ1,2=-p (x )±p 2(x )-4q (x )2㊂因为x (∈1+1-2aττ,1+1-δa δ-)τ,0<δ<τ,所以p 2(x )-4q (x )=δ(2x 2(-1+1-aτ))(τx 2(-1+1+2aτ))τ<0,q (x )-1=δ(δ-τ)(τx 2(-1+1-δa δ-))τ>0,从而有点Z (x ,y )的特征值为一对共轭复数λ1,2=-p (x )2±i4q (x )-p 2(x )2,且‖λ1,2‖=q (x )>1㊂(ⅱ)为选取适当的r x 0和r y 0,先证明1+1-2aττ<3(1-a )<1+1-δaδ-τ㊂因为a <1,τ(2-3a )+12(1-a )<δ,所以τ(2-3a )+1<2δ(1-a ),从而δ-τ+1-δa <3(1-a )(δ-τ)㊂又δ<τ,故3(1-a )<1+1-δa δ-τ(5)由a >23,0<τ<12-a -22a (1-a ),可得(3a -2)τ-2aτ+1<3a -22-a -22a (1-a )-2aτ+1<3a -22-a -243×13-2aτ+1=-2-2aτ<0,从而有1+1-2aττ-3(1-a )=(3a -2)τ-2aτ+1τ<0,即1+1-2aττ<3(1-a )(6)由式(5)㊁(6)可知1+1-2aττ<3(1-a )<1+1-δa δ-τ取0<r x 0{<min 1+1-δaδ-τ-3(1-a ),3(1-a )-1+1-2aτ}τ,r y 0∈R +,考虑不动点Z 0(x 0,y 0)=(3(1-a ),a 3(1-a )+b )的邻域U Z 0{=(x ,y )|(x -x 0)2r 2x 0+(y -y 0)2r 2y 0≤}1,任取点Z (x ,y )∈U Z 0,则其横坐标x ∈[x 0-r x 0,x 0+r x 0],显然[x 0-r x 0,x 0+r x 0](⊂1+1-2aττ,1+1-δaδ-)τ,故U Z 0⊂U ㊂命题得证㊂引理2 考虑不动点Z 0(x 0,y 0)=(3(1-a ),a3(1-a )+b ),及其邻域U Z 0{=(x ,y )|(x -x 0)2r 2x 0+(y -y 0)2r 2y 0≤1,r x 0=0.1,r y 0∈R }+,若式(2)~(4),以及δ≠τ1+aτ(7)30x 0+100(δ2x 0(-33a +1+4δ+4δaτ-))δ(㊃δ2aτ(τ-δ)2)(-1+1<3003a -2+1δ+δaτ-)δ(8)成立,则在邻域U Z 0中存在一点Z *(x *,y *),满足Z *≠Z 0,f 2(Z *)=Z 0,且|D f 2(Z *)|≠0㊂证明:设有点Z *(x *,y *),令Z 1(x 1,y 1)=f (Z *),f 2(Z *)=f (Z 1)=Z 0,即x 1=x *+(δx *-(x *)33-y *+)I y 1=y *+δτ(ax *-y *+b ìîíïïïï)(9)且㊃121㊃第2期 陈晓楠等:一类离散神经系统中的混沌存在性x 1+(δx 1-x 313-y 1+)I=xy 1+δτ(ax 1-y 1+b )=y {(10)易知方程组(10)的一组解为x 1+=-x 0(+33a +1+4δ+4δa τ-)δ2y 1+=y 0-δa (x 1+-x 0)τ-{δ(11)先证(x 1+,y 1+)≠(x 0,y 0)㊂由a >23,τ>δ可(得2x 0(-33a +1+4δ+4δaτ-))(δ-2x-(33a +1+4δ+4δa τ-))δ(=37a -3+4δ+4δa τ-)δ>(37×23-3+4δ+4δa τ-)δ(=5+121δ+δaτ-)δ>0,从而x 0-x 1+=2x 0(-33a +1+4δ+4δaτ-)δ≠0(12)将式(11)代入到方程组(10)可得(x *)3-3(δ1+δ+δ2a τ-)δx*+3(δ1-δ2aτ(τ-δ))2x 1++3aτ2(τ-δ)2x 0=0(13)y *=τ(τ-δ)y 0-δaτ(x 1+-x 0)-δ(ax *+b )(τ-δ)(τ-δ)2(14)为求方程(13)在[x 0-0.1,x 0+0.1]内的解x *,令x *=s +x 0,代入方程(13)可得g (s )≡s 3+3s 2x 0+3(s x 20-1-1δ-δaτ-)δ+3δ(×δ2aτ(τ-δ)2)-1(x 0-x 1+)=0(15)下面证明方程(15)在[-0.1,0.1]内有非零解,即g (s )有非零的零点㊂令g′(s )(=3s 2+2x 0s +x 20-1-1δ-δaδ-)τ=0,可得g (s )的两个驻点s 1,2=-x 0±1+1δ+δaτ-δ,其中s 1<s 2,显然当s ∈(s 1,s 2)时,g′(s )<0㊂因为τ>δ>0,所以1+1-δa δ-τ=1+δaτ-δ-1τ-δ<1+δa τ-δ+1δ,从而有s 2=1+1δ+δa τ-δ-3(1-a )>1+1-δaδ-τ-3(1-a )>0.1,且s 1=-s 2-23(1-a )<-0.1,故[-0.1,0.1]⊂(s 1,s 2)㊂因此当s ∈[-0.1,0.1]时,g′(s )<0,即g (s )在区间[-0.1,0.1]是单调递减函数㊂由式(7)及aτ>1,有δ2aτ-(τ-δ)2=(δ(1+aτ)-τ)(δ(aτ-1)+τ)≠0(16)由式(12)㊁(16)及τ>δ>0可得g (0)=3(x 0-x 1+)(δ2aτ-(τ-δ)2)δ(τ-δ)2≠0㊂且由式(8)㊁(12)可知,g (-0.1)g (0.1)(=30(1000x 0+100(x 0-x 1+)(δδ2aτ(τ-δ)2)))-12(-1(1000(3003a -2+1δ+δaτ-)δ))-12<0,根据零点定理得,∃s *∈(-0.1,0.1),g (s *)=0,又因g (0)≠0,故s *≠0㊂综上方程(13)有实根x *=x 0+s *∈(x 0-0.1,x 0+0.1),且x *≠x 0㊂取r y 0>|y *-y 0|,从而Z *(x *,y *)∈U Z 0,Z *≠Z 0,f 2(Z *)=Z 0㊂下证|D f 2(Z *)|≠0㊂|D f (Z *)|>0|D f (Z 1)|=δ2a τ(+1-δ)τ(1+δ(1-x21+))=(-1-δ)æèççτ3δ(1-a ()3a +1+4δ+4δa τ-)δ4+3δa2+2+öø÷÷δ-2δ2a τ<0故|D f 2(Z *)|=|D f (Z 1)‖D f (Z *)|≠0㊂命题得证㊂说明:引理2中的r x 0取值为0.1,事实上只要r x 0取充分小的正数,补充形如式(8)的条件,即可得相同的结论㊂综上所述,可得如下定理㊂定理 若系统参数a ㊁τ㊁δ满足引理1与引理2中的条件,则系统(1)的不动点Z 0(x 0,y 0)=(3(1-a ),a3(1-a )+b )是snap⁃back repel⁃ler,此时系统(1)是Marotto 混沌的,即存在:1)一个正整数N ,使得对于任一整数p ≥N ,f 有㊃221㊃北京化工大学学报(自然科学版) 2023年周期p 点㊂2)一个 scrambled set”,即一个不含周期点的不可数集S 满足(a)f [S ]⊂S ;(b)对于S 中任意不相同的两点X 和Y ,有lim k →∞sup ‖f k (X )-f k (Y )‖>0;(c)对于S 中任意点X ,以及f 的任意周期点Y ,有lim k →∞sup ‖f k (X )-f k (Y )‖>0㊂3)S 有一个不可数子集S 0,对于S 0中任意点X和Y ,有lim k →∞inf ‖f k (X )-f k (Y )‖=0㊂2 数值模拟本节取两组参数,通过数值计算,对Marotto 混沌吸引子及周期轨等复杂动态进行模拟㊂1)参数组1:b =I =1,a =0.9,τ=2.47,δ=0.992㊂系统(1)存在一个不动点㊂Z 0(x 0,y 0)=(0.54772255750517,1.492950301754653),其特征值为λ1,2=1.14639±0.24135i,‖λ1,2‖>1㊂此时参数满足引理1与引理2的条件:a =0.9>23τ(2-3a )+12(1-a )=-3.645<δ<11.143=2a -13a -2<τ<12-a -22a (1-a )=3.977δ≠τ1+aτ=0.9915830x 0+100(δ2x 0(-33a +1+4δ+4δaτ-))δ(×δ2aτ(τ-δ)2)(-1+1=3.488<3003a -2+1δ+δa τ-)δ=693.637引理1中的点集U 与Z 0的邻域U Z 0分别为U {=(x ,y )|x (∈1+1-2aττ,1+1-δa δ-)τ=(0.4445,0.9631),y ∈}R U Z 0{=(x ,y )|(x -x 0)20.01+(y -y 0)24≤}1显然U Z 0⊂U ㊂∀Z (x ,y )∈U Z 0,Z (x ,y )的特征方程为λ2+p (x )λ+q (x )=0,特征值为λ1,2=-p (x )±p 2(x )-4q (x )2,‖λ1,2‖=q (x )其中p 2(x )-4q (x )<p 2(x 0-0.1)-4q (x 0-0.1)=-0.00679<0,q (x )-1>q (x 0+0.1)-1=0.44606>0,即Z 的特征值为一对模长大于1的共轭复数㊂U Z 0中存在一点Z *(x *,y *)=(0.54652802940618,-0.462001800839748),Z 1(x 1,y 1)=f (Z *)=(2.485010260145796,0.32271318340719),f 2(Z *)=Z 0,且|D f 2(Z *)|=-2.9045㊂由定理可得Z 0是一个snap⁃back repel⁃ler,即系统(1)存在Marotto 混沌吸引子㊂选取不动点Z 0(x 0,y 0)=(0.54772255750517,1.492950301754653)的邻域U Z 0中的点(0.5477,1.493)为初始点,其相图见图1,这条轨道的Lya⁃punov 指数为{-0.24197,0.07866},即为Marotto混沌吸引子㊂图1 Marotto 混沌吸引子(δ=0.992)Fig.1 A Marotto chaotic attractor(δ=0.992)不同于前一组Z 0的邻域U Z 0中r x 0(=0.1)的取值,将r x 0设置为0.08,仍能证明在一定参数条件下系统是Marotto 意义下混沌的㊂2)参数组2:b =I =1,a =0.91,τ=2.4,δ=0.99㊂系统(1)存在不动点㊂Z 0(x 0,y 0)=(0.519615242270663,1.472849870466304),其特征值为λ1,2=1.1551±0.2334i,‖λ1,2‖>1㊂存在Z 0的邻域U Z 0{=(x ,y )|(x -x 0)2r 2x 0+(y -y 0)2r 2y 0≤1,r x 0=0.08,r y 0}=2.2任取Z (x ,y )∈U Z 0,因为p 2(x )-4q (x )<p 2(x 0-㊃321㊃第2期 陈晓楠等:一类离散神经系统中的混沌存在性0.08)-4q (x 0-0.08)=-0.01955<0,q (x )-1>q (x 0+0.08)=0.33163>0,所以Z (x ,y )有一对模长大于1的共轭复特征值-p (x )±i4q (x )-p 2(x )2,且其模长q (x )>1㊂U Z 0中存在点Z *(x *,y *)=(0.45525855258397,0.664929193926399),Z 1(x 1,y 1)=f (Z *)=(2.523106596180992,0.19274677774448),f 2(Z *)=Z 0,|D f 2(Z *)|=-3.0704≠0,由条件①㊁②得,Z 0(x 0,y 0)是snap⁃back repeller,系统(1)存在Marotto 混沌吸引子,这条轨道的Lyapunov 指数为{-0.18254,0.0651}㊂针对第一组参数值,在δ=0.992时系统(1)出现了Marotto 混沌吸引子㊂本文计算了参数δ∈(0.9,1)时,以不动点Z 0邻域中的点(0.5477,1.493)为初始点的轨道的最大Lyapunov 指数,如图2所示㊂很明显,系统在δ参数区间(0.988,1)内始终保持混沌状态㊂图2 最大Lyapunov 指数图Fig.2 The maximumLyapunov exponent diagram通过数值模拟计算,我们还发现系统存在高周期轨道,如周期16轨㊁周期14轨,如图3㊁4所示㊂图3 周期16轨(δ=0.9205)Fig.3 Period⁃16orbit(δ=0.9205)图4 周期14轨(δ=0.962)Fig.4 Period⁃14orbit(δ=0.962)3 结束语本文研究了一类离散FHN 神经系统的混沌现象,主要从理论上严格证明了该系统Marotto 意义下混沌的存在性㊂此外通过数值计算模拟出Marotto 混沌吸引子,同时还发现系统中存在周期14轨和周期16轨,这些复杂动态的发现丰富了对此系统的理解㊂参考文献:[1] 曲良辉,都琳,胡海威,等.电磁刺激对FHN 神经元系统的调控作用[J].动力学与控制学报,2020,18(1):40-48.QU L H,DU L,HU H W,et al.Regulation of electro⁃magnetic stimulation on FHN neuronal system[J].Jour⁃nal of Dynamics and Control,2020,18(1):40-48.(in Chinese)[2] 武春艳.神经元FitzHugh⁃Nagumo 模型的动力学分析[D].太原:太原理工大学,2015.WU C Y.Dynamical analysis of the neuronal FitzHugh⁃Nagumo model [D ].Taiyuan:Taiyuan University of Technolgy,2015.(in Chinese)[3] HODGKIN A L,HUXLEY A F.A quantitative descrip⁃tion of membrane current and its application to conductionand excitation in nerve[J].Journal of Physiology,1952,117(4):500-544.[4] FITZHUGH R.Impulses and physiological states in theo⁃retical models of nerve membrane[J].Biophysical Jour⁃nal,1961,1(6):445-466.[5] NAGUMO J,ARIMOTO S,YOSHIZAWA S.An activepulse transmission line simulating nerve axon [J].Pro⁃ceedings of the IRE,1962,50(10):2061-2070.[6] HAYASHI M.A note on the uniqueness of the closed or⁃bit of the FitzHugh -Nagumo system [J].Quarterly ofApplied Mathematics,2000,58(1):171-176.㊃421㊃北京化工大学学报(自然科学版) 2023年[7] ROCŞOREANU C,GIURGIŢEANU N,GEORGESCUA.Connections between saddles for the FitzHugh -Nagu⁃mo system [J].International Journal of Bifurcation and Chaos,2001,11(2):533-540.[8] JING Z J,JIA Z Y,CHANG Y.Chaos behavior in thediscrete FitzHugh nerve system [J].Science in China (Series A),2001,44(12):1571-1578.[9] ZHAN F B,LIU S Q.A Hénon⁃like map inspired by thegeneralized discrete⁃time FitzHugh -Nagumo model[J].Nonlinear Dynamics,2019,97(10):2675-2691.[10]MAROTTO F R.Snap⁃back repellers imply chaos in R n [J].Journal of Mathematical Analysis and Applicaions,1978,63(1):199-223.[11]MAROTTO F R.On redefining a snap⁃back repeller[J].Chaos,Solitions and Fractals,2005,25(1):25-28.Chaos in a discrete FitzHugh -Nagumo modelCHEN XiaoNan CHANG Yu * CHEN Jiao YANG RuiFeng(College of Mathematics and Physics,Beijing University of Chemical Technology,Beijing 100029,China)Abstract :Chaos phenomena in a discrete FitzHugh -Nagumo nervous model have been investigated.The existence of chaos in the sense of Marotto is analyzed theoretically.Numerical calculation studies for Lyapunov exponent dem⁃onstrate various complex dynamics,such as chaotic attractors,and period⁃14,and period⁃16orbits.Key words :discrete FitzHugh -Nagumo model;chaos in the sense of Marotto;Lyapunov exponent(责任编辑:吴万玲)㊃521㊃第2期 陈晓楠等:一类离散神经系统中的混沌存在性。
上海财经大学计量经济学试卷
诚实考试吾心不虚 , 公平竞争方显实力, 考试失败尚有机会 , 考试舞弊前功尽弃。
上海财经大学《计量经济学 》课程考试卷（A ）闭卷课程代码 课程序号2008—2009 学年第 1 学期姓名 学号 班级一、单选题(每小题2分, 共计40分)1.如果模型中变量在10%的显著性水平下是显著的, 则（ D ）A. 该变量在5%的显著性水平下是也显著的B. 该变量在1%和5%的显著性水平下都是显著的C. 如果P 值为12%, 则该变量在15%的显著性水平下也是显著的 D 、 如果P 值为2%, 则该变量在5%的显著性水平下也是显著的 2.高斯－马尔可夫是( D )A.摇滚乐.B.足球运动.C.鲜美的菜.D.估计理论中的著名定理, 来自于著名的统计学家: Johan.Car.Friedric.Gauss 和Andre.Andreevic.Markov 。
3.以下关于工具变量的说法不正确的是（ B ）。
A.与随机干扰项不相关B. 与所替代的随机解释变量不相关C.与所替代的随机解释变量高度相关D.与模型中其他解释变量不相关4.在含有截距项的多元回归中, 校正的判定系数 与判定系数R2的关系有: （ B ） A.R2＜ B.R2＞ C.R2＝ D.R2与 的关系不能确定5.根据样本资料估计得出人均消费支出Y 对人均收入X 的回归模型为lnYi=2.00+0.75lnXi+ei, 这表明人均收入每增加1%, 人均消费支出将大约增加（ B ）A.0.2..B.0.75..C.2..D.7.5%6.在存在异方差的情况下, 普通最小二乘法（OLS ）估计量是（ B ） A.有偏的, 非有效的 B.无偏的, 非有效的 C.有偏的, 有效的 D.无偏的, 有效的7.已知模型的普通最小二乘法估计残差的一阶自相关系数为0, 则DW 统计量的近似值为（ C ）A.0B.1C.2D.48.在多元回归模型中, 若某个解释变量对其余解释变量回归后的判定系数接近1, 则表明原模型中存在（C）A.异方差性B.自相关C.多重共线性D.拟合优度低9.设某商品需求模型为: Yi＝β0＋β1Xi＋Ui, 其中Y是商品的需求量, X是商品的价格, 为了考虑全年12个月份季节变动的影响, 假设模型中引入了12个虚拟变量, 则会产生的问题是（D）A.异方差性B.自相关C.不完全的多重共线性D.完全的多重共线性10.下列表述不正确的是（D）A.尽管存在不完全的多重共线性, 普通最小二乘估计量仍然是最优线性无偏估计量B.双对数模型的R2可以与对数－线性模型的R2相比较, 但不能与线性－对数模型的R2相比较。
STD函数的内部计算公式
STD函数的内部计算公式各股票软件的标准差函数STD是不同的，⽽布林线的上下轨是以STD为基础计算出来的，所以使⽤布林线应⼩⼼。
以2008/3/28的上证综指为例，利⽤如下代码："收盘价3⽇STD:STD(CLOSE,3);"，三⽇收盘价分别是：3606.86，3580.15，3411.49，在飞狐交易师中显⽰的3⽇收盘价标准差是105.928，⼤智慧新⼀代中显⽰的是105.932，通达信中是86.49，同花顺中显⽰74.90。
⽤EXCEL中的函数STDEV计算的样本⽅差是105.9316，STDEVP计算总体⽅差是86.49。
可见⼤智慧和飞狐使⽤的算法是样本⽅差，⼆者数据基本⼀致，⼤智慧的更精确⼀点，通达信使⽤的是总体⽅差，同花顺就不知所谓了。
这⾥最关键的是要明⽩总体⽅差与样本⽅差的区别。
总体⽅差(population variance) ：如果这组数据本⾝便构成⼀个总体, 均差平⽅和除以数据中观察值的数⽬，称为总体⽅差。
如⼀组数据X1,X2,..., Xn：其平均值M=(X1+X2+...+Xn)/n ，总体⽅差为 [ (X1-M)^2+...(Xn-M)^2 ]/n 的平⽅根。
对于⽆限总体，N为⽆限⼤。
样本是由总体中任意抽取⽽形成的，样本的各种数量关系(包括平均值和⽅差)都是总体的相关数量的估计值。
数理统计学已经证明了，对于从总体中抽取的样本，⽤前⾯的总体⽅式公式计算出来的⽅差值来估计总体的⽅差总是偏⼩的。
样本⽅差有时也称为样本均⽅（mean square, 简记为MS），是总体⽅差的⽆偏估计，计算公式是[ (X1-M)^2+...(Xn-M)^2 ]/(n-1)的平⽅根，也就是说⽤(n-1)取代n作为分母。
为什么⽤n-1⽽不是n呢？这可以从⾃由度来解释。
这样看，X1,X2,...Xn是n个可以⾃由变化的样本，互不影响。
⽽X1-M, X2-M,...Xn-M是否也是n个⾃由变化的呢？不是……因为这n个统计量受到⼀个约束条件的影响就是之和等于0。
FDA最新指导原则：药物分析程序及方法验证指导原则
Analytical Procedures and Methods Validation for Drugsand BiologicsDRAFT GUIDANCEThis guidance document is being distributed for comment purposes only. Comments and suggestions regarding this draft document should be submitted within 90 days of publication in the Federal Register of the notice announcing the availability of the draft guidance. Submit electronic comments to . Submit written comments to the Division of Dockets Management (HFA-305), Food and Drug Administration, 5630 Fishers Lane, rm. 1061, Rockville, MD 20852. All comments should be identified with the docket number listed in the notice of availability that publishes in the Federal Registe r.For questions regarding this draft document contact (CDER) Lucinda Buhse 314-539-2134, or (CBER) Office of Communication, Outreach and Development at 800-835-4709 or 301-827-1800.U.S. Department of Health and Human ServicesFood and Drug AdministrationCenter for Drug Evaluation and Research (CDER)Center for Biologics Evaluation and Research (CBER)February 2014CMCAnalytical Procedures and Methods Validation for Drugsand BiologicsAdditional copies are available from:Office of CommunicationsDivision of Drug Information, WO51, Room 2201Center for Drug Evaluation and ResearchFood and Drug Administration10903 New Hampshire Ave., Silver Spring, MD 20993Phone: 301-796-3400; Fax: 301-847-8714druginfo@/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/default.htmand/orOffice of Communication, Outreach andDevelopment, HFM-40Center for Biologics Evaluation and ResearchFood and Drug Administration1401 Rockville Pike, Rockville, MD 20852-1448ocod@/BiologicsBloodVaccines/GuidanceComplianceRegulatoryInformation/Guidances/default.htm(Tel) 800-835-4709 or 301-827-1800U.S. Department of Health and Human ServicesFood and Drug AdministrationCenter for Drug Evaluation and Research (CDER)Center for Biologics Evaluation and Research (CBER)Febr uary 2014CMCTABLE OF CONTENTSI.INTRODUCTION (1)II.BACKGROUND (2)III.ANALYTICAL METHODS DEVELOPMENT (3)IV.CONTENT OF ANALYTICAL PROCEDURES (3)A.Principle/Scope (4)B.Apparatus/Equipment (4)C.Operating Parameters (4)D.Reagents/Standards (4)E.Sample Preparation (4)F.Standards Control Solution Preparation (5)G.Procedure (5)H.System Suitability (5)I.Calculations (5)J.Data Reporting (5)V.REFERENCE STANDARDS AND MATERIALS (6)VI.ANALYTICAL METHOD VALIDATION FOR NDA, ANDAs, BLAs, AND DMFs (6)A.Noncompendial Analytical Procedures (6)B.Validation Characteristics (7)pendial Analytical Procedures (8)VII.STATISTICAL ANALYSIS AND MODELS (8)A.Statistics (8)B.Models (8)VIII.LIFE CYCLE MANAGEMENT OF ANALYTICAL PROCEDURES (9)A.Revalidation (9)B.Analytical Method Comparability Studies (10)1.Alternative Analytical Procedures (10)2.Analytical Methods Transfer Studies (11)C.Reporting Postmarketing Changes to an Approved NDA, ANDA, or BLA (11)IX.FDA METHODS VERIFICATION (12)X.REFERENCES (12)Guidance for Industry11Analytical Procedures and Methods Validation for Drugs and2Biologics345This draft guidance, when finalized, will represent the Food and Drug Administration’s (FDA’s) current 6thinking on this topic. It does not create or confer any rights for or on any person and does not operate to 7bind FDA or the public. You can use an alternative approach if the approach satisfies the requirements of 8the applicable statutes and regulations. If you want to discuss an alternative approach, contact the FDA9staff responsible for implementing this guidance. If you cannot identify the appropriate FDA staff, call 10the appropriate number listed on the title page of this guidance.11121314I. INTRODUCTION1516This revised draft guidance supersedes the 2000 draft guidance for industry on Analytical17Procedures and Methods Validation2,3 and, when finalized, will also replace the 1987 FDA18guidance for industry on Submitting Samples and Analytical Data for Methods Validation. It19provides recommendations on how you, the applicant, can submit analytical procedures4 and20methods validation data to support the documentation of the identity, strength, quality, purity,21and potency of drug substances and drug products.5It will help you assemble information and 22present data to support your analytical methodologies. The recommendations apply to drug23substances and drug products covered in new drug applications (NDAs), abbreviated new drug 24applications (ANDAs), biologics license applications (BLAs), and supplements to these25applications. The principles in this revised draft guidance also apply to drug substances and drug 26products covered in Type II drug master files (DMFs).2728This revised draft guidance complements the International Conference on Harmonisation (ICH) 29guidance Q2(R1)Validation of Analytical Procedures: Text and Methodology(Q2(R1)) for30developing and validating analytical methods.3132This revised draft guidance does not address investigational new drug application (IND) methods 33validation, but sponsors preparing INDs should consider the recommendations in this guidance.34For INDs, sufficient information is required at each phase of an investigation to ensure proper35identity, quality, purity, strength, and/or potency. The amount of information on analytical36procedures and methods validation will vary with the phase of the investigation.6 For general371 This guidance has been prepared by the Office of Pharmaceutical Science, in the Center for Drug Evaluation andResearch (CDER) and the Center for Biologics Evaluation and Research (CBER) at the Food and DrugAdministration.2 Sample submission is described in section IX, FDA Methods Verification.3 We update guidances periodically. To make sure you have the most recent version of a guidance, check the FDADrugs guidance Web page at/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/default.htm.4Analytical procedure is interchangeable with a method or test procedure.5The terms drug substance and drug product, as used in this guidance, refer to human drugs and biologics.6 See 21 CFR 312.23(a)(7).guidance on analytical procedures and methods validation information to be submitted for phase 38one studies, sponsors should refer to the FDA guidance for industry on Content and Format of39Investigational New Drug Applications (INDs) for Phase 1 Studies of Drugs, Including40Well-Characterized, Therapeutic, Biotechnology-Derived Products. General considerations for 41analytical procedures and method validation (e.g., bioassay) before conduct of phase three42studies are discussed in the FDA guidance for industry on IND Meetings for Human Drugs and 43Biologics, Chemistry, Manufacturing, and Controls Information.4445This revised draft guidance does not address specific method validation recommendations for46biological and immunochemical assays for characterization and quality control of many drug47substances and drug products. For example, some bioassays are based on animal challenge48models, and immunogenicity assessments or other immunoassays have unique features that49should be considered during development and validation.5051In addition, the need for revalidation of existing analytical methods may need to be considered 52when the manufacturing process changes during the product’s life cycle. For questions on53appropriate validation approaches for analytical procedures or submission of information not54addressed in this guidance, you should consult with the appropriate FDA product quality review 55staff.5657If you choose a different approach than those recommended in this revised draft guidance, we58encourage you to discuss the matter with the appropriate FDA product quality review staff before 59you submit your application.6061FDA’s guidance documents, including this guidance, do not establish legally enforceable62responsibilities. Instead, guidances describe the Agency’s current thinking on a topic and should 63be viewed only as recommendations, unless specific regulatory or statutory requirements are64cited. The use of the word should in Agency guidances means that something is suggested or65recommended, but not required.666768II.BACKGROUND6970Each NDA and ANDA must include the analytical procedures necessary to ensure the identity, 71strength, quality, purity, and potency of the drug substance and drug product.7 Each BLA must 72include a full description of the manufacturing methods, including analytical procedures that73demonstrate the manufactured product meets prescribed standards of identity, quality, safety,74purity, and potency.8 Data must be available to establish that the analytical procedures used in 75testing meet proper standards of accuracy and reliability and are suitable for their intended76purpose.9 For BLAs and their supplements, the analytical procedures and their validation are77submitted as part of license applications or supplements and are evaluated by FDA quality78review groups.79807 See 21 CFR 314.50(d)(1) and 314.94(a)(9)(i).8 See 21 CFR 601.2(a) and 601.2(c).9 See 21 CFR 211.165(e) and 211.194(a)(2).Analytical procedures and validation data should be submitted in the corresponding sections of 81the application in the ICH M2 eCTD: Electronic Common Technical Document Specification.108283When an analytical procedure is approved/licensed as part of the NDA, ANDA, or BLA, it84becomes the FDA approved analytical procedure for the approved product. This analytical85procedure may originate from FDA recognized sources (e.g., a compendial procedure from the 86United States Pharmacopeia/National Formulary (USP/NF)) or a validated procedure you87submitted that was determined to be acceptable by FDA. To apply an analytical method to a88different product, appropriate validation studies with the matrix of the new product should be89considered.909192III.ANALYTICAL METHODS DEVELOPMENT9394An analytical procedure is developed to test a defined characteristic of the drug substance or95drug product against established acceptance criteria for that characteristic. Early in the96development of a new analytical procedure, the choice of analytical instrumentation and97methodology should be selected based on the intended purpose and scope of the analytical98method. Parameters that may be evaluated during method development are specificity, linearity, 99limits of detection (LOD) and quantitation limits (LOQ), range, accuracy, and precision.100101During early stages of method development, the robustness of methods should be evaluated102because this characteristic can help you decide which method you will submit for approval.103Analytical procedures in the early stages of development are initially developed based on a104combination of mechanistic understanding of the basic methodology and prior experience.105Experimental data from early procedures can be used to guide further development. You should 106submit development data within the method validation section if they support the validation of 107the method.108109To fully understand the effect of changes in method parameters on an analytical procedure, you 110should adopt a systematic approach for method robustness study (e.g., a design of experiments 111with method parameters). You should begin with an initial risk assessment and follow with112multivariate experiments. Such approaches allow you to understand factorial parameter effects 113on method performance. Evaluation of a method’s performance may include analyses of114samples obtained from in-process manufacturing stages to the finished product. Knowledge115gained during these studies on the sources of method variation can help you assess the method 116performance.117118119IV.CONTENT OF ANALYTICAL PROCEDURES120121You should describe analytical procedures in sufficient detail to allow a competent analyst to 122reproduce the necessary conditions and obtain results within the proposed acceptance criteria. 123You should also describe aspects of the analytical procedures that require special attention. An 124analytical procedure may be referenced from FDA recognized sources (e.g., USP/NF,12510 See sections 3.2.S.4 Control of Drug Substance, 3.2.P.4 Control of Excipients, and 3.2.P.5 Control of DrugProduct.Association of Analytical Communities (AOAC) International)11 if the referenced analytical126procedure is not modified beyond what is allowed in the published method. You should provide 127in detail the procedures from other published sources. The following is a list of essential128information you should include for an analytical procedure:129130A.Principle/Scope131132A description of the basic principles of the analytical test/technology (separation, detection, etc.); 133target analyte(s) and sample(s) type (e.g., drug substance, drug product, impurities or compounds 134in biological fluids, etc.).135136B.Apparatus/Equipment137138All required qualified equipment and components (e.g., instrument type, detector, column type, 139dimensions, and alternative column, filter type, etc.).140141C.Operating Parameters142143Qualified optimal settings and ranges (allowed adjustments) critical to the analysis (e.g., flow144rate, components temperatures, run time, detector settings, gradient, head space sampler). A145drawing with experimental configuration and integration parameters may be used, as applicable. 146147D.Reagents/Standards148149The following should be listed:150151•Grade of chemical (e.g., USP/NF, American Chemical Society, High152Performance or Pressure Liquid Chromatography, or Gas153Chromatography and preservative free).154•Source (e.g., USP reference standard or qualified in-house reference material). 155•State (e.g., dried, undried, etc.) and concentration.156•Standard potencies (purity correction factors).157•Storage controls.158•Directions for safe use (as per current Safety Data Sheet).159•Validated or useable shelf life.160161New batches of biological reagents, such as monoclonal antibodies, polyclonal antisera, or cells, 162may need extensive qualification procedures included as part of the analytical procedure.163164E.Sample Preparation165166Procedures (e.g., extraction method, dilution or concentration, desalting procedures and mixing 167by sonication, shaking or sonication time, etc.) for the preparations for individual sample tests. 168A single preparation for qualitative and replicate preparations for quantitative tests with16911 See 21 CFR 211.194(a)(2).appropriate units of concentrations for working solutions (e.g., µg/ml or mg/ml) and information 170on stability of solutions and storage conditions.171172F.Standards Control Solution Preparation173174Procedures for the preparation and use of all standard and control solutions with appropriate175units of concentration and information on stability of standards and storage conditions,176including calibration standards, internal standards, system suitability standards, etc.177178G.Procedure179180A step-by-step description of the method (e.g., equilibration times, and scan/injection sequence 181with blanks, placeboes, samples, controls, sensitivity solution (for impurity method) and182standards to maintain validity of the system suitability during the span of analysis) and allowable 183operating ranges and adjustments if applicable.184185H.System Suitability186187Confirmatory test(s) procedures and parameters to ensure that the system (equipment,188electronics, and analytical operations and controls to be analyzed) will function correctly as an 189integrated system at the time of use. The system suitability acceptance criteria applied to190standards and controls, such as peak tailing, precision and resolution acceptance criteria, may be 191required as applicable. For system suitability of chromatographic systems, refer to CDER192reviewer guidance on Validation of Chromatographic Methods and USP General Chapter <621> 193Chromatography.194195I.Calculations196197The integration method and representative calculation formulas for data analysis (standards,198controls, samples) for tests based on label claim and specification (e.g., assay, specified and199unspecified impurities and relative response factors). This includes a description of any200mathematical transformations or formulas used in data analysis, along with a scientific201justification for any correction factors used.202203J.Data Reporting204205A presentation of numeric data that is consistent with instrumental capabilities and acceptance 206criteria. The method should indicate what format to use to report results (e.g., percentage label 207claim, weight/weight, and weight/volume etc.) with the specific number of significant figures 208needed. The American Society for Testing and Materials (ASTM) E29 describes a standard209practice for using significant digits in test data to determine conformance with specifications. For 210chromatographic methods, you should include retention times (RTs) for identification with211reference standard comparison basis, relative retention times (RRTs) (known and unknown212impurities) acceptable ranges and sample results reporting criteria.213214215V.REFERENCE STANDARDS AND MATERIALS216217Primary and secondary reference standards and materials are defined and discussed in the218following ICH guidances: Q6A Specifications: Test Procedures and Acceptance Criteria for 219New Drug Substances and New Drug Products: Chemical Substances (ICH Q6A), Q6B220Specifications: Test Procedures and Acceptance Criteria for Biotechnological/Biological221Products, and Q7 Good Manufacturing Practice Guidance for Active Pharmaceutical222Ingredients. For all standards, you should ensure the suitability for use. Reference standards for 223drug substances are particularly critical in validating specificity for an identity test. You should 224strictly follow storage, usage conditions, and handling instructions for reference standards to225avoid added impurities and inaccurate analysis. For biological products, you should include226information supporting any reference standards and materials that you intend to use in the BLA 227and in subsequent annual reports for subsequent reference standard qualifications. Information 228supporting reference standards and materials include qualification test protocols, reports, and 229certificates of analysis (including stability protocols and relevant known impurity profile230information, as applicable).231232Reference standards can often be obtained from USP and may also be available through the233European Pharmacopoeia, Japanese Pharmacopoeia, World Health Organization, or National 234Institute of Standards and Technology. Reference standards for a number of biological products 235are also available from CBER. For certain biological products marketed in the U.S., reference 236standards authorized by CBER must be used before the product can be released to the market.12 237Reference materials from other sources should be characterized by procedures including routine 238and beyond routine release testing as described in ICH Q6A. You should consider orthogonal 239methods. Additional testing could include attributes to determine the suitability of the reference 240material not necessarily captured by the drug substance or product release tests (e.g., more241extensive structural identity and orthogonal techniques for purity and impurities, biological242activity).243244For biological reference standards and materials, we recommend that you follow a two-tiered 245approach when qualifying new reference standards to help prevent drift in the quality attributes 246and provide a long-term link to clinical trial material. A two-tiered approach involves a247comparison of each new working reference standard with a primary reference standard so that it 248is linked to clinical trial material and the current manufacturing process.249250251VI.ANALYTICAL METHOD VALIDATION FOR NDA, ANDAs, BLAs, AND 252DMFs253254A.Noncompendial Analytical Procedures255256Analytical method validation is the process of demonstrating that an analytical procedure is257suitable for its intended purpose. The methodology and objective of the analytical procedures 258should be clearly defined and understood before initiating validation studies. This understanding 25912 See 21 CFR 610.20.is obtained from scientifically-based method development and optimization studies. Validation 260data must be generated under an protocol approved by the sponsor following current good261manufacturing practices with the description of methodology of each characteristic test and262predetermined and justified acceptance criteria, using qualified instrumentation operated under 263current good manufacturing practices conditions.13 Protocols for both drug substance and264product analytes or mixture of analytes in respective matrices should be developed and executed. 265266ICH Q2(R1) is considered the primary reference for recommendations and definitions on267validation characteristics for analytical procedures. The FDA Reviewer Guidance: Validation of 268Chromatographic Methods is available as well.269270B.Validation Characteristics271272Although not all of the validation characteristics are applicable for all types of tests, typical273validation characteristics are:274275•Specificity276•Linearity277•Accuracy278•Precision (repeatability, intermediate precision, and reproducibility)279•Range280•Quantitation limit281•Detection limit282283If a procedure is a validated quantitative analytical procedure that can detect changes in a quality 284attribute(s) of the drug substance and drug product during storage, it is considered a stability285indicating assay. To demonstrate specificity of a stability-indicating assay, a combination of286challenges should be performed. Some challenges include the use of samples spiked with target 287analytes and all known interferences; samples that have undergone various laboratory stress288conditions; and actual product samples (produced by the final manufacturing process) that are289either aged or have been stored under accelerated temperature and humidity conditions.290291As the holder of the NDA, ANDA, or BLA, you must:14 (1) submit the data used to establish292that the analytical procedures used in testing meet proper standards of accuracy and reliability, 293and (2) notify the FDA about each change in each condition established in an approved294application beyond the variations already provided for in the application, including changes to 295analytical procedures and other established controls.296297The submitted data should include the results from the robustness evaluation of the method,298which is typically conducted during method development or as part of a planned validation299study.1530013 See 21 CFR 211.165(e); 21 CFR 314.50 (d), and for biologics see 21 CFR 601.2(a), 601.2(c), and 601.12(a).14 For drugs see 21 CFR 314.50 (d), 314.70(d), and for biologics see 21 CFR 601.2(a), 601.2(c), and 601.12(a). For aBLA, as discussed below, you must obtain prior approval from FDA before implementing a change in analyticalmethods if those methods are specified in FDA regulations15 See section III and ICH Q2(R1).pendial Analytical Procedures302303The suitability of an analytical procedure (e.g., USP/NF, the AOAC International Book of304Methods, or other recognized standard references) should be verified under actual conditions of 305use.16 Compendial general chapters, which are complex and mention multiple steps and/or306address multiple techniques, should be rationalized for the intended use and verified. Information 307to demonstrate that USP/NF analytical procedures are suitable for the drug product or drug308substance should be included in the submission and generated under a verification protocol.309310The verification protocol should include, but is not limited to: (1) compendial methodology to 311be verified with predetermined acceptance criteria, and (2) details of the methodology (e.g.,312suitability of reagent(s), equipment, component(s), chromatographic conditions, column, detector 313type(s), sensitivity of detector signal response, system suitability, sample preparation and314stability). The procedure and extent of verification should dictate which validation characteristic 315tests should be included in the protocol (e.g., specificity, LOD, LOQ, precision, accuracy, etc.). 316Considerations that may influence what characteristic tests should be in the protocol may depend 317on situations such as whether specification limits are set tighter than compendial acceptance318criteria, or RT or RRT profiles are changing in chromatographic methods because of the319synthetic route of drug substance or differences in manufacturing process or matrix of drug320product. Robustness studies of compendial assays do not need to be included, if methods are 321followed without deviations.322323324VII.STATISTICAL ANALYSIS AND MODELS325326A.Statistics327328Statistical analysis of validation data can be used to evaluate validation characteristics against 329predetermined acceptance criteria. All statistical procedures and parameters used in the analysis 330of the data should be based on sound principles and appropriate for the intended evaluation.331Reportable statistics of linear regression analysis R (correlation coefficient), R square332(coefficient of determination), slope, least square, analysis of variance (ANOVA), confidence 333intervals, etc., should be provided with justification.For information on statistical techniques 334used in making comparisons, as well as other general information on the interpretation and335treatment of analytical data, appropriate literature or texts should be consulted.17336337B.Models338339Some analytical methods might use chemometric and/or multivariate models. When developing 340these models, you should include a statistically adequate number and range of samples for model 341development and comparable samples for model validation. Suitable software should be used for 342data analysis. Model parameters should be deliberately varied to test model robustness.34334416 See 21 CFR 211.194(a)(2) and USP General Chapter <1226> Verification of Compendial Procedures.17 See References section for examples including USP <1010> Analytical Data – Interpretation and Treatment.。