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Lecture 22: Bivariate Normal Distribution Distribution
6.5 Conditional Distributions General Bivariate Normal Let Z1; Z2 ∼ N (0; 1), which we will use to build a general bivariate normal Lecture 22: Bivariate Normal Distribution distribution. 1 1 2 2 f (z1; z2) = exp − (z1 + z2 ) Statistics 104 2π 2 We want to transform these unit normal distributions to have the follow Colin Rundel arbitrary parameters: µX ; µY ; σX ; σY ; ρ April 11, 2012 X = σX Z1 + µX p 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 1 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - Marginals General Bivariate Normal - Cov/Corr First, lets examine the marginal distributions of X and Y , Second, we can find Cov(X ; Y ) and ρ(X ; Y ) Cov(X ; Y ) = E [(X − E(X ))(Y − E(Y ))] X = σX Z1 + µX h p i = E (σ Z + µ − µ )(σ [ρZ + 1 − ρ2Z ] + µ − µ ) = σX N (0; 1) + µX X 1 X X Y 1 2 Y Y 2 h p 2 i = N (µX ; σX ) = E (σX Z1)(σY [ρZ1 + 1 − ρ Z2]) h 2 p 2 i = σX σY E ρZ1 + 1 − ρ Z1Z2 p 2 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY = σX σY ρE[Z1 ] p 2 = σX σY ρ = σY [ρN (0; 1) + 1 − ρ N (0; 1)] + µY = σ [N (0; ρ2) + N (0; 1 − ρ2)] + µ Y Y Cov(X ; Y ) ρ(X ; Y ) = = ρ = σY N (0; 1) + µY σX σY 2 = N (µY ; σY ) Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 2 / 22 Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 3 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - RNG Multivariate Change of Variables Consequently, if we want to generate a Bivariate Normal random variable Let X1;:::; Xn have a continuous joint distribution with pdf f defined of S. -
Applied Biostatistics Mean and Standard Deviation the Mean the Median Is Not the Only Measure of Central Value for a Distribution
Health Sciences M.Sc. Programme Applied Biostatistics Mean and Standard Deviation The mean The median is not the only measure of central value for a distribution. Another is the arithmetic mean or average, usually referred to simply as the mean. This is found by taking the sum of the observations and dividing by their number. The mean is often denoted by a little bar over the symbol for the variable, e.g. x . The sample mean has much nicer mathematical properties than the median and is thus more useful for the comparison methods described later. The median is a very useful descriptive statistic, but not much used for other purposes. Median, mean and skewness The sum of the 57 FEV1s is 231.51 and hence the mean is 231.51/57 = 4.06. This is very close to the median, 4.1, so the median is within 1% of the mean. This is not so for the triglyceride data. The median triglyceride is 0.46 but the mean is 0.51, which is higher. The median is 10% away from the mean. If the distribution is symmetrical the sample mean and median will be about the same, but in a skew distribution they will not. If the distribution is skew to the right, as for serum triglyceride, the mean will be greater, if it is skew to the left the median will be greater. This is because the values in the tails affect the mean but not the median. Figure 1 shows the positions of the mean and median on the histogram of triglyceride. -
1. How Different Is the T Distribution from the Normal?
Statistics 101–106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M §7.1 and §7.2, ignoring starred parts. Reread M&M §3.2. The eects of estimated variances on normal approximations. t-distributions. Comparison of two means: pooling of estimates of variances, or paired observations. In Lecture 6, when discussing comparison of two Binomial proportions, I was content to estimate unknown variances when calculating statistics that were to be treated as approximately normally distributed. You might have worried about the effect of variability of the estimate. W. S. Gosset (“Student”) considered a similar problem in a very famous 1908 paper, where the role of Student’s t-distribution was first recognized. Gosset discovered that the effect of estimated variances could be described exactly in a simplified problem where n independent observations X1,...,Xn are taken from (, ) = ( + ...+ )/ a normal√ distribution, N . The sample mean, X X1 Xn n has a N(, / n) distribution. The random variable X Z = √ / n 2 2 Phas a standard normal distribution. If we estimate by the sample variance, s = ( )2/( ) i Xi X n 1 , then the resulting statistic, X T = √ s/ n no longer has a normal distribution. It has a t-distribution on n 1 degrees of freedom. Remark. I have written T , instead of the t used by M&M page 505. I find it causes confusion that t refers to both the name of the statistic and the name of its distribution. As you will soon see, the estimation of the variance has the effect of spreading out the distribution a little beyond what it would be if were used. -
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a specific time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we define (i) Def. Distribution function: (fi-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an infinite matrix). -
Spatial Autocorrelation: Covariance and Semivariance Semivariance
Spatial Autocorrelation: Covariance and Semivariancence Lily Housese P eters GEOG 593 November 10, 2009 Quantitative Terrain Descriptorsrs Covariance and Semivariogram areare numericc methods used to describe the character of the terrain (ex. Terrain roughness, irregularity) Terrain character has important implications for: 1. the type of sampling strategy chosen 2. estimating DTM accuracy (after sampling and reconstruction) Spatial Autocorrelationon The First Law of Geography ““ Everything is related to everything else, but near things are moo re related than distant things.” (Waldo Tobler) The value of a variable at one point in space is related to the value of that same variable in a nearby location Ex. Moranan ’s I, Gearyary ’s C, LISA Positive Spatial Autocorrelation (Neighbors are similar) Negative Spatial Autocorrelation (Neighbors are dissimilar) R(d) = correlation coefficient of all the points with horizontal interval (d) Covariance The degree of similarity between pairs of surface points The value of similarity is an indicator of the complexity of the terrain surface Smaller the similarity = more complex the terrain surface V = Variance calculated from all N points Cov (d) = Covariance of all points with horizontal interval d Z i = Height of point i M = average height of all points Z i+d = elevation of the point with an interval of d from i Semivariancee Expresses the degree of relationship between points on a surface Equal to half the variance of the differences between all possible points spaced a constant distance apart -
Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis
Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis Hervé Cardot, Antoine Godichon-Baggioni Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, 9, rue Alain Savary, 21078 Dijon, France July 12, 2016 Abstract The geometric median covariation matrix is a robust multivariate indicator of dis- persion which can be extended without any difficulty to functional data. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high dimensional data without being obliged to store all the data in memory. Asymptotic convergence prop- erties of the recursive algorithms are studied under weak conditions. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high dimension data. An illustration on a large sample and high dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 hours confirms the interest of consider- ing the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN. Keywords. Averaging, Functional data, Geometric median, Online algorithms, Online principal components, Recursive robust estimation, Stochastic gradient, Weiszfeld’s algo- arXiv:1504.02852v5 [math.ST] 9 Jul 2016 rithm. -
Linear Regression
eesc BC 3017 statistics notes 1 LINEAR REGRESSION Systematic var iation in the true value Up to now, wehav e been thinking about measurement as sampling of values from an ensemble of all possible outcomes in order to estimate the true value (which would, according to our previous discussion, be well approximated by the mean of a very large sample). Givenasample of outcomes, we have sometimes checked the hypothesis that it is a random sample from some ensemble of outcomes, by plotting the data points against some other variable, such as ordinal position. Under the hypothesis of random selection, no clear trend should appear.Howev er, the contrary case, where one finds a clear trend, is very important. Aclear trend can be a discovery,rather than a nuisance! Whether it is adiscovery or a nuisance (or both) depends on what one finds out about the reasons underlying the trend. In either case one must be prepared to deal with trends in analyzing data. Figure 2.1 (a) shows a plot of (hypothetical) data in which there is a very clear trend. The yaxis scales concentration of coliform bacteria sampled from rivers in various regions (units are colonies per liter). The x axis is a hypothetical indexofregional urbanization, ranging from 1 to 10. The hypothetical data consist of 6 different measurements at each levelofurbanization. The mean of each set of 6 measurements givesarough estimate of the true value for coliform bacteria concentration for rivers in a region with that urbanization level. The jagged dark line drawn on the graph connects these estimates of true value and makes the trend quite clear: more extensive urbanization is associated with higher true values of bacteria concentration. -
Covariance of Cross-Correlations: Towards Efficient Measures for Large-Scale Structure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Mon. Not. R. Astron. Soc. 400, 851–865 (2009) doi:10.1111/j.1365-2966.2009.15490.x Covariance of cross-correlations: towards efficient measures for large-scale structure Robert E. Smith Institute for Theoretical Physics, University of Zurich, Zurich CH 8037, Switzerland Accepted 2009 August 4. Received 2009 July 17; in original form 2009 June 13 ABSTRACT We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multitracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show that for the usual autopower statistic, besides the off-diagonal covariance generated through gravitational mode- coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as spatial frequencies become larger than the fundamental mode of the survey volume. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (S/N), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, the cross-power approach can outperform the autopower one by factors of a few. -
On the Use of the Autocorrelation and Covariance Methods for Feedforward Control of Transverse Angle and Position Jitter in Linear Particle Beam Accelerators*
'^C 7 ON THE USE OF THE AUTOCORRELATION AND COVARIANCE METHODS FOR FEEDFORWARD CONTROL OF TRANSVERSE ANGLE AND POSITION JITTER IN LINEAR PARTICLE BEAM ACCELERATORS* Dean S. Ban- Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439 ABSTRACT It is desired to design a predictive feedforward transverse jitter control system to control both angle and position jitter m pulsed linear accelerators. Such a system will increase the accuracy and bandwidth of correction over that of currently available feedback correction systems. Intrapulse correction is performed. An offline process actually "learns" the properties of the jitter, and uses these properties to apply correction to the beam. The correction weights calculated offline are downloaded to a real-time analog correction system between macropulses. Jitter data were taken at the Los Alamos National Laboratory (LANL) Ground Test Accelerator (GTA) telescope experiment at Argonne National Laboratory (ANL). The experiment consisted of the LANL telescope connected to the ANL ZGS proton source and linac. A simulation of the correction system using this data was shown to decrease the average rms jitter by a factor of two over that of a comparable standard feedback correction system. The system also improved the correction bandwidth. INTRODUCTION Figure 1 shows the standard setup for a feedforward transverse jitter control system. Note that one pickup #1 and two kickers are needed to correct beam position jitter, while two pickup #l's and one kicker are needed to correct beam trajectory-angle jitter. pickup #1 kicker pickup #2 Beam Fast loop Slow loop Processor Figure 1. Feedforward Transverse Jitter Control System It is assumed that the beam is fast enough to beat the correction signal (through the fast loop) to the kicker. -
Lecture 4 Multivariate Normal Distribution and Multivariate CLT
Lecture 4 Multivariate normal distribution and multivariate CLT. T We start with several simple observations. If X = (x1; : : : ; xk) is a k 1 random vector then its expectation is × T EX = (Ex1; : : : ; Exk) and its covariance matrix is Cov(X) = E(X EX)(X EX)T : − − Notice that a covariance matrix is always symmetric Cov(X)T = Cov(X) and nonnegative definite, i.e. for any k 1 vector a, × a T Cov(X)a = Ea T (X EX)(X EX)T a T = E a T (X EX) 2 0: − − j − j � We will often use that for any vector X its squared length can be written as X 2 = XT X: If we multiply a random k 1 vector X by a n k matrix A then the covariancej j of Y = AX is a n n matrix × × × Cov(Y ) = EA(X EX)(X EX)T AT = ACov(X)AT : − − T Multivariate normal distribution. Let us consider a k 1 vector g = (g1; : : : ; gk) of i.i.d. standard normal random variables. The covariance of g is,× obviously, a k k identity × matrix, Cov(g) = I: Given a n k matrix A, the covariance of Ag is a n n matrix × × � := Cov(Ag) = AIAT = AAT : Definition. The distribution of a vector Ag is called a (multivariate) normal distribution with covariance � and is denoted N(0; �): One can also shift this disrtibution, the distribution of Ag + a is called a normal distri bution with mean a and covariance � and is denoted N(a; �): There is one potential problem 23 with the above definition - we assume that the distribution depends only on covariance ma trix � and does not depend on the construction, i.e. -
Lecture 2 Covariance Is the Statistical Measure That Indicates The
Lecture 2 Covariance is the statistical measure that indicates the interactive risk of a security relative to others in a portfolio of securities. In other words, the way security returns vary with each other affects the overall risk of the portfolio. The covariance between two securities X and Y may be calculated using the following formula: Where: Covxy = Covariance between x and y. Rx = Return of security x. Ry = Return of security y Rx = Expected or mean return of security x. Ry = Expected or mean return of security y. N = Number of observations. Calculation of Covariance Year Rx Deviation R…3 Deviation Product of deviations Rx – Rx y Ry – Ry (Rx – Rx) (Ry – Ry) 1 10 -4 17 5 -20 2 12 -2 13 1 -2 3 16 2 10 -2 -4 4 18 4 8 -4 -16 = -42 / 4 = -10.5 The covariance is a measure of how returns of two securities move together. If the returns of the two securities move in the same direction consistently the covariance would be positive. If the returns of the two securities move in opposite direction consistently the covariance would be negative. If the movements of returns are independent of each other, covariance would be close to zero. Covariance is an absolute measure of interactive risk between two securities. To facilitate comparison, covariance can be standardized. Dividing the covariance between two securities by product of the standard deviation of each security gives such a standardised measure. This measure is called the coefficient of correlation. This may be expressed as: Where Rxy = Coefficient of correlation between x and y Covxy = Covariance between x and y. -
The Central Limit Theorem (Review)
Introduction to Confidence Intervals { Solutions STAT-UB.0103 { Statistics for Business Control and Regression Models The Central Limit Theorem (Review) 1. You draw a random sample of size n = 64 from a population with mean µ = 50 and standard deviation σ = 16. From this, you compute the sample mean, X¯. (a) What are the expectation and standard deviation of X¯? Solution: E[X¯] = µ = 50; σ 16 sd[X¯] = p = p = 2: n 64 (b) Approximately what is the probability that the sample mean is above 54? Solution: The sample mean has expectation 50 and standard deviation 2. By the central limit theorem, the sample mean is approximately normally distributed. Thus, by the empirical rule, there is roughly a 2.5% chance of being above 54 (2 standard deviations above the mean). (c) Do you need any additional assumptions for part (c) to be true? Solution: No. Since the sample size is large (n ≥ 30), the central limit theorem applies. 2. You draw a random sample of size n = 16 from a population with mean µ = 100 and standard deviation σ = 20. From this, you compute the sample mean, X¯. (a) What are the expectation and standard deviation of X¯? Solution: E[X¯] = µ = 100; σ 20 sd[X¯] = p = p = 5: n 16 (b) Approximately what is the probability that the sample mean is between 95 and 105? Solution: The sample mean has expectation 100 and standard deviation 5. If it is approximately normal, then we can use the empirical rule to say that there is a 68% of being between 95 and 105 (within one standard deviation of its expecation).