Paleo-ecological temperature estimationExperimental growth-T relationshipsProblem: there are more than 30 planktonic foraminifera speciesErikson faunal vs. Emiliani isotopic records from the Caribbean SeaHow does one deal with complete faunal counts?Variance and CovarianceCorrelation CoefficientLinear regressionLinear regression 1Numerical exampleLinear Regression 3Correlation in n dimensions Ellipsoids and eigenvectorsEigenvectors 2Matrix collection of eigenvectorsSquare orthonormalPrinciple Components Analysis and Factor AnalysisSpecies Space and Sample SpaceWhat is sample space?Species as vectors in sample space:Correlation coefficient as the angle between species in sample spaceFactor analysis 2Singular value decompositionProduct moment and singular valuesFactor analysis - problemsImbrie-Kipp (1971) method: factor analysis and transfer functionsImbrie-Kipp 1Imbrie-Kipp 2Imbrie-Kipp 3Imbrie-Kipp 4Imbrie-Kipp 5Imbrie and Kipp (1971) foraminiferal factors vs. temperatureImbrie and Kipp (1971) factors vs. T, regression fitImbrie-Kipp 6Imbrie-Kipp 6Imbrie-Kipp 7Imbrie-Kipp 8Imbrie-Kipp 9Critique of the Imbrie-Kipp Method 1Critique of the Imbrie-Kipp Method 2Critique of the Imbrie-Kipp Method 3An Alternative to the Imbrie-Kipp method: Modern Analogue Technique (MAT) A paleoclimate conundrumPrell et al. (1976) : Factor analyzing tropical glacial age samples turns up a factor that doesn’t appear in the core top sampCLIMAP redone after downcore factors addedNo-analogue factors?ReadingsMIT OpenCourseWare http://ocw.mit.edu 12.740 Paleoceanography Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Paleo-ecological temperature estimation12.740 Lecture 4 Spring 2008Paleo-ecological temperature estimationPaleo-ecological temperature estimation is based on the empirical observation that certain species of marine organisms live and leave fossils on the seafloor that correspond to temperature patterns in oceanic surface waters. Experimental manipulations of organisms show that many have preferred temperature ranges. Downcorerecords confirm known glacial/interglacial fluctuations.Experimental growth-T relationshipsfrom Wolfgang Berger108642010 203040 50Growth Rate (Doublings / day)Maximum growth rateTemperature OCGrowth rates of some species of phytoplankton as a function of temperature. At optimum conditions for growth, warm-adapted plankton have a greater rate of production than do cold-adapted ones. Curves based on responses of certain diatoms and chlorophytes (left to right Detonula confervacea. Dit ylum brightwellii. Dunaliella tertiolecta. Chlorella pyrenoidosa).Figure by MIT OpenCourseWare.Problem: there are more than 30 planktonic foraminifera species• You have to randomly split sediment samples down to 300-600 specimens and then examine them under a binocular microscope. Sometimes this requires rolling the specimen over. Even today, computer-aided recognition isn’t up to the task.• So the method of counting all forams is slow and tedious.• To get around this problem, Erikson developed methods based on single indicator species,e.g. N. pachyderma (L) or G. menardii.Erikson faunal vs. Emiliani isotopic records from the Caribbean SeaImage removed due to copyright restrictions.Please see : Imbrie and Imbrie (1979) Ice Ages: Solving the Mystery, Figure 33.How does one deal with complete faunal counts?• As you have just seen, plotting each species percentages vs depth results in a visually tedious presentation whose meaning is hard to grasp.• John Imbrie’s solution: statistical factor analysisVariance and Covariance• Variance (of x is denoted Sx2; variance of y is denoted Sy2) is a measure of the scatter of values of a variable about its mean:• Covariance (of x and y) expresses the relationship between two variables (a measure of the scatter of values of points in a plane relative to the centroid of the data set):Sx2= (xi− x)2i=1n∑/ nSxy2= (xi− x)i=1n∑(yi− y)/nCorrelation Coefficientwhere sx= std. dev. of x = [Σ(xi-x)2 / (n-1)]1/2and sy= std. dev. of y = [Σ(yi-y)2/ (n-1)]1/2•sxand syrelate the deviations of points from the average relative to the "range" (actually std. dev.) of the observations.•r2is "the variance of Y accounted for by its covariance with x" (usually expressed in % units).• In other words, r2is the Covariance divided by the Variance.r =1n −1(xi− xsxi=1n∑)(yi− ysy) =1n −1(xi− x)(yi− y)i=1n∑sxsyLinear regression1. Common linear regression of y on x: y = A + BxLet S = Σ (yi- A - B xi)2To minimize S, set ∂S/∂A =0; ∂S/∂B =0; solve for A and B.2. Matrix math solution of Linear Regression:for eq'n Ax = b (m eq'ns, n unknowns),if columns of A are linearly independent,then:x = (ATA)−1ATbregression 1For example, for the simple linear regression y = C + Dx, where we want to fit pairs of data xi, yiwe want to find C, D that mimimizeΣ [yi-(C + Dxi)]2In matrix form, we write the equation y = C + Dxas:1 x1y11 x2C y2. . D = .. . .1 xnyni.e. A x = bNumerical exampleNumerical example:1 1 4.11 2 C 5.91 3 D = 7.81 4 10.31 5 12.1gives C=1.92, D=2.04x = (ATA)−1ATbLinear Regression 3Similarly, to solve the equation y = A + Bx +Cx2:1 x1x12y11 x2x22Ay2. . B = . . . C .1 xnxn2ynSimple matrix formulas also allow you to compute the estimated uncertainties of the regression coefficients and the correlation coefficients.Correlation in n dimensions• Multiple linear regression e.g. y = A + Bx1+ Cx2(the equation for a plane in 3D space)• r-matrix (later we will sometimes refer to this as the matrix Σ)Property 1 2 3 4 51 1.00 0.86 0.45 0.83 0.452 0.86 1.00 0.74 0.23 0.643 0.45 0.74 1.00 0.78 0.57(ρij)4 0.83 0.23 0.78 1.00 0.395 0.45 0.64 0.57 0.39 1.001 x1-1x2-1y11 x1-2x2-2Ay2. . B = . . . C .1 x1-nx2-nynEllipsoids and eigenvectorsA x = λ xFigure by MIT OpenCourseWare.Adapted from source: Joreskog et al.Geological Factor Analysis (1976).square eigen- eigen- eigen-matrix vector value vectorOne way to find eigenvectors:1steigenvector = major axis of ellipsoid2ndeigenvector = largest minor axis of ellipsoid…etc.This works more or less as if we did a regression to get a line that "explains" most of the variance(the dominant linear trend of the data in n-dimensional space), subtracted that regression from the data, then perform