sundermeyer mar 999 spring 2009 1 laboratory in oceanography: data and methods mar599, spring 2009...

SundermeyerMAR 999

Spring 2009 1

Laboratory in Oceanography: Data and Methods

MAR599, Spring 2009

Miles A. Sundermeyer

Intro to the Statistics Toolbox

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Spring 2009 2

Measures of Central Tendency

• Geometric Mean:

• Harmonic Mean:

Intro to Statistics ToolboxStatistics Toolbox/Descriptive Statistics

Function Name Description

Geomean Geometric mean

harmmean Harmonic mean

mean Arithmetic mean

median 50th percentile

mode Most frequent value

trimmean Trimmed mean (specify percentile)

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Measures of Dispersion

• Interquartile range: difference between the 75th and 25th percentiles

• Mean absolute deviation: mean(abs(x-mean(x)))

• Moment: mean((x-mean(x)).^order (e.g., order=2 gives variance)

• skewness: third central moment of x, divided by cube of its standard deviation (pos/neg skewness implies longer right/left tail)

• kurtosis: fourth central moment of x, divided by 4th power of its standard deviation (high kurtosis means sharper peak and longer/fatter tails)



irq Interquartile range

mad Mean absolute deviation

moment Central moment of all orders

range Range

std Standard deviation

var Variance

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Examples of Skewness & Kurtosis:

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Bootstrap Method

• Involves choosing random samples with replacement from a data set and analyzing each sample data set the same way as the original data set. The number of elements in each bootstrap sample set equals the number of elements in the original data set. The range of sample estimates obtained provides a means of estimating uncertainty of the quantity being estimated.

• In general, bootstrap method can be used to compute uncertainty for any functional calculation, provided the sample data set is ‘representative’ of the true distribution.

Jacknife Method

• Similar to the bootstrap is the jackknife, but uses re-sampling to estimate the bias and variance of sample statistics.


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Example:Bootstrap Method for estimating uncertainty on Lagrangian Integral Time Scale (from Sundermeyer and Price, 1998)

“Integrating the LACFs using 100 days as the upper limit of the integral of Rii() in (12) gives the integral timescales I(11,22) = (10.6 ± 4.8, 5.4 ± 2.8) days for the (zonal, meridional) components, where uncertainties represent 95% confidence limits estimated using a bootstrap method [e.g., Press et al., 1986].”

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Intro to Statistics ToolboxStatistics Toolbox/Statistical Visualization

Probability Distribution Plots

• Normal Probability Plots:>> x = normrnd(10,1,25,1);>> normplot(x)

>> x = exprnd(10,100,1);>> normplot(x)

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• Quantile-Quantile Plots:>> x = poissrnd(10,50,1); y = poissrnd(5,100,1);>> qqplot(x,y);

>> x = normrnd(5,1,100,1); >> y = wblrnd(2,0.5,100,1);>> qqplot(x,y);

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• Cumulative Distribution Plots:>> y = evrnd(0,3,100,1);>> cdfplot(y)>> hold on>> x = -20:0.1:10;>> f = evcdf(x,0,3);>> plot(x,f,'m')>> legend('Empirical', ...

'Theoretical', ...'Location','NW')

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Intro to Statistics ToolboxStatistics Toolbox/Probability Distributions/Supported Distributions

Supported distributions include wide range of:

• Continuous distributions (data)

• Continuous distributions (statistics)

• Discrete distributions

• Multivariate distributions

http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/&http://www.mathworks.com/support/product/product.html?product=ST


pdf Probability density functions

cdf Cumulative distribution functions

inv Inverse cumulative distribution functions

stat Distribution statistics functions

fit Distribution fitting functions

like Negative log-likelihood functions

rnd Random number generators

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Supported distributions (cont’d)

Name pdf cdf inv stat fit like rnd

...

Normal (Gaussian)

Normpdf, pdf

Normcdf, cdf

norminv, icdf

normstatnormfit, mle, dfittool

normlike

normrnd, randn, random, randtool

Pearson system

pearsrnd pearsrnd

Piecewise pdf cdf icdf paretotails random

Rayleighraylpdf, pdf

raylcdf, cdf

raylinv, icdf

raylstatraylfit, mle, dfittool

raylrnd, random, randtool

...

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Name pdf cdf inv stat fit like rnd

Chi-square chi2pdf, pdf

chi2cdf, cdf

chi2inv, icdf

chi2stat

chi2rnd, random, randtool

F fpdf, pdf fcdf, cdf finv, icdf fstat frnd, random, randtool

Noncentral chi-square ncx2pdf, pdf

ncx2cdf, cdf

ncx2inv, icdf

ncx2stat

ncx2rnd, random, randtool

Noncentral F ncfpdf, pdf

ncfcdf, cdf

ncfinv, icdf

ncfstat ncfrnd, random, randtool

Noncentral t nctpdf, pdf

nctcdf, cdf

nctinv, icdf

nctstat nctrnd, random, randtool

Student's t tpdf, pdf tcdf, cdf tinv, icdf tstat trnd, random, randtool

t location- scale dfittool


Supported statistics

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Intro to Statistics ToolboxStatistics Toolbox/Hypothesis Tests

Hypothesis Testing

• Can only disprove a hypothesis

• null hypothesis – an assertion about a population. It is "null" in that it represents a status quo belief, such as the absence of a characteristic or the lack of an effect.

• alternative hypothesis – a contrasting assertion about the population that can be tested against the null hypothesis

H1: µ ≠ null hypothesis value — (two-tailed test)H1: µ > null hypothesis value — (right-tail test)H1: µ< null hypothesis value — (left-tail test)

• test statistic – random sample of population collected, and test statistic computed to characterize the sample. The statistic varies with type of test, but distribution under null hypothesis must be known (or assumed).

• p-value - probability, under null hypothesis, of obtaining a value of the test statistic as extreme or more extreme than the value computed from the sample.

• significance level - threshold of probability, typical value of is 0.05. If p-value < the test rejects the null hypothesis; if p-value > α, there is insufficient evidence to reject the null hypothesis.

• confidence interval - estimated range of values with a specified probability of containing the true population value of a parameter.

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Hypothesis Testing

• Hypothesis tests make assumptions about the distribution of the random variable being sampled in the data. These must be considered when choosing a test and when interpreting the results.

• Z-test (ztest) and the t-test (ttest) both assume that the data are independently sampled from a normal distribution.

• Both the z-test and the t-test are relatively robust with respect to departures from this assumption, so long as the sample size n is large enough.

• Difference between the z-test and the t-test is in the assumption of the standard deviation σ of the underlying normal distribution. A z-test assumes that σ is known; a t-test does not. Thus t-test must determine from the sample.

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ztest

• The test requires σ (the standard deviation of the population) to be known

• The formula for calculating the z score for the z-test is:

where:x is the sample mean

μ is the mean of the population

• The z-score is compared to a z-table, which contains the percent of area under the normal curve between the mean and the z-score. This table will indicate whether the calculated z-score is within the realm of chance, or if it is so different from the mean that the sample mean is unlikely to have happened by chance.

n

xz

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http://www.stats4students.com/Essentials/Standard-Score/Overview.php

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ttest

• Like z-test, except the t-test does not require σ to be known

• The formula for calculating the t score for the t-test is:

where:x is the sample mean

μ is the mean of the populations is the sample variance

• Under the null hypothesis that the population is distributed with mean μ, the z-statistic has a standard normal distribution, N(0,1). Under the same null hypothesis, the t-statistic has Student's t distribution with n – 1 degrees of freedom.

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http://www.stats4students.com/Essentials/Standard-Score/Overview.php

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ttest2

• performs a t-test of the null hypothesis that data in the vectors x and y are independent random samples from normal distributions with equal means and equal but unknown variances – unknown variances may be either equal or unequal.

• The formula for calculating the score for the t-test2 is:

where:x, y are sample means

sx, sy are the sample variances

• The null hypothesis is that the two samples are distributed with the same mean.

http://www.socialresearchmethods.net/kb/stat_t.php


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Function Description

ansaribradley Ansari-Bradley test. Tests if two independent samples come from the same distribution, against the alternative that they come from distributions that have the same median and shape but different variances.

chi2gof Chi-square goodness-of-fit test. Tests if a sample comes from a specified distribution, against the alternative that it does not come from that distribution.

dwtest Durbin-Watson test. Tests if the residuals from a linear regression are independent, against the alternative that there is autocorrelation among them.

jbtest Jarque-Bera test. Tests if a sample comes from a normal distribution with unknown mean and variance, against the alternative that it does not come from a normal distribution.

linhyptest Linear hypothesis test. Tests if H*b = c for parameter estimates b with estimated covariance H and specified c, against the alternative that H*b ≠ c.

kstest One-sample Kolmogorov-Smirnov test. Tests if a sample comes from a continuous distribution with specified parameters, against the alternative that it does not come from that distribution.

kstest2 Two-sample Kolmogorov-Smirnov test. Tests if two samples come from the same continuous distribution, against the alternative that they do not come from the same distribution.

lillietest Lilliefors test. Tests if a sample comes from a distribution in the normal family, against the alternative that it does not come from a normal distribution.

ranksum Wilcoxon rank sum test. Tests if two independent samples come from identical continuous distributions with equal medians, against the alternative that they do not have equal medians.

runstest Runs test. Tests if a sequence of values comes in random order, against the alternative that the ordering is not random.

signrank One-sample or paired-sample Wilcoxon signed rank test. Tests if a sample comes from a continuous distribution symmetric about a specified median, against the alternative that it does not have that median.

signtest One-sample or paired-sample sign test. Tests if a sample comes from an arbitrary continuous distribution with a specified median, against the alternative that it does not have that median.

ttest One-sample or paired-sample t-test. Tests if a sample comes from a normal distribution with unknown variance and a specified mean, against the alternative that it does not have that mean.

ttest2 Two-sample t-test. Tests if two independent samples come from normal distributions with unknown but equal (or, optionally, unequal) variances and the same mean, against the alternative that the means are unequal.

vartest One-sample chi-square variance test. Tests if a sample comes from a normal distribution with specified variance, against the alternative that it comes from a normal distribution with a different variance.

vartest2 Two-sample F-test for equal variances. Tests if two independent samples come from normal distributions with the same variance, against the alternative that they come from normal distributions with different variances.

vartestn Bartlett multiple-sample test for equal variances. Tests if multiple samples come from normal distributions with the same variance, against the alternative that they come from normal distributions with different variances.

ztest One-sample z-test. Tests if a sample comes from a normal distribution with known variance and specified mean, against the alternative that it does not have that mean.

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ANOVA (ANalysis Of VAriance)

• ANOVA is like a t-test among multiple (typically >2) data sets simultaneously

• T-tests can be done between two data sets, or one set and a “true” value

• uses the f-distribution instead of the t-distribution

• assumes that all of the data sets have equal variances

One-way ANOVA is a simple special case of the linear model. The one-way ANOVA form of the model is

where: • yij is a matrix of observations, each column represents a different group.

• .j is a matrix whose columns are the group means. (The "dot j" notation means applies to all rows of column j. That is, αij is the same for all i.)

• εij is a matrix of random disturbances.

The model assumes that the columns of y are a constant plus a random disturbance. ANOVA tests if the constants are all the same.

Intro to Statistics ToolboxStatistics Toolbox/Analysis of Variance

ijjijy .

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One-way ANOVA

Example: Hogg and Ledolter bacteria counts in milk. Columns represent different shipments, rows are bacteria counts from cartons chosen randomly from each shipment. Do some shipments have higher counts than others?

>> load hogg>> hogg

hogg =24 14 11 7 19

15 7 9 7 24 21 12 7 4 19 27 17 13 7 15 33 14 12 12 10 23 16 18 18 20

>> [p,tbl,stats] = anova1(hogg);>> p

p = 1.1971e-04

• standard ANOVA table has columns for the sums of squares, dof, mean squares (SS/df), F statistic, and p-value.

• P-value is from F statistic of hypothesis test whether bacteria counts are same.


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One-way ANOVA (cont’d)• In this case the p-value is about 0.0001, a very small value. This is a strong indication

that the bacteria counts from the different shipments are not the same. An F statistic as extreme as this would occur by chance only once in 10,000 times if the counts were truly equal.

• The p-value returned by anova1 depends on assumptions about random disturbances εij in the model equation. For the p-value to be correct, these disturbances need to be: independent, normally distributed, and have constant variance.


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Multiple Comparisons• Sometimes need to determine not just whether there are differences among

means, but which pairs of means are significantly different.

• In t-test, compute t-statistic and compare to a critical value. However, when testing multiple pairs, for example, if probability of t-statistic exceeding critical value is 5%, then for 10 pairs, much more likely that one of these will falsely fail that criterion.

• Can perform a multiple comparison test using the multcompare function by supplying it with the stats output from anova1.

Example:>> load hogg>> [p,tbl,stats] = anova1(hogg);>> [c,m] = multcompare(stats)

Example:see Light_DO.m


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Two-way ANOVA Determine whether data from several groups have a common mean. Differs from one-

way ANOVA in that the groups in two-way ANOVA have two categories of defining characteristics instead of one (e.g., think of two independent variables/dimensions)

Two-way ANOVA is again a special case of the linear model. The two-way ANOVA form of the model is

where: • yijk is a matrix of observations (with rows i, columns j, and repetition k).• is a constant matrix of the overall mean of the observations.• .j is a matrix whose columns are deviations of each observation attributable to the

first independent variable. All values in a given column of are identical, and values in each row sum to 0.

• .j is a matrix whose rows are the deviations of each observation attributable to the second independent variable. All values in a given row of are identical, and values in each column of sum to 0.

• ij is a matrix of interactions. Values in each row sum to 0, and values in each column sum to 0.

• εij is a matrix of random disturbances.

The model assumes that the columns of y are a series of constants plus a random disturbance. You want to know if the constants are all the same.


ijkijjjijky ..

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Two-way ANOVA

Example: Determine effect of car model and factory on the mileage rating of cars.There are three models (columns) and two factories (rows). Data from first factory is in

first three rows, data from second factory is in last three rows. Do some cars have different mileage than others?

>> load mileagemileage =33.3000 34.5000 37.400033.4000 34.8000 36.800032.9000 33.8000 37.600032.6000 33.4000 36.600032.5000 33.7000 37.000033.0000 33.9000 36.7000

>> cars = 3;>> [p,tbl,stats] = anova2(mileage,cars);[p,tbl,stats] = anova1(hogg);


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Two-way ANOVA (cont’d)• In this case the p-value for the first effect is zero to four decimal places. This

indicates that the effect of the first predictor varies from one sample to another. An F statistic as extreme as this would occur by chance only once in 10,000 times if the samples were truly equal.

• The p-value for the second effect is 0.0039, which is also highly significant. This indicates that the effect of the second predictor varies from one sample to another.

• Does not appear to be any interaction between the two predictors. The p-value, 0.8411, means that the observed result is quite likely (84 out 100 times) given that

there is no interaction.

• The p-values returned by anova2 depend on assumptions about the random disturbances εij in the model equation. For the p-values to be correct, these disturbances need to be: independent, normally distributed, and have constant variance.

• In addition, anova2 requires that data be balanced, which means there must be the same number of samples for each combination of control variables. Other ANOVA methods support unbalanced data with any number of predictors.


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Linear Regression Models• In statistics, linear regression models take the form of a summation of

coefficient · (independent variable or combination of independent variables).

For example:

• In this example, the response variable y is modeled as a combination of constant, linear, interaction, and quadratic terms formed from two predictor variables x1 and x2.

• Uncontrolled factors and experimental errors are modeled by ε. Given data on x1, x2, and y, regression estimates the model parameters βj (j = 1, ..., 5).

• More general linear regression models represent the relationship between a continuous response y and a continuous or categorical predictor x in the form:

Intro to Statistics ToolboxStatistics Toolbox/Regression Analysis

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Example (system of equations):

Suppose we have a series of measurements of stream discharge and stage, measured at n different times.

time (day) = [0 14 28 42 56 70] stage (m) = [0.612 0.647 0.580 0.629 0.688 0.583]discharge (m3/s) = [0.330 0.395 0.241 0.338 0.531 0.279]

Suppose we now wish to fit a rating curve to these measurements. Let x = stage, y = discharge, then we can write this series of measurements as:

yi = mxi + b, with i = 1:n.

This in turn can be written as: y = X b, or:

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yi = mxi + b

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1) (cos) sin(

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Example: Harmonic Analysis:• sin(q+f) = sin()cos() + sin(cos()

• Let: A=Ccos(), B=Csin()=> Csin(t+) = Asin(t) + Bcos(t)

• Linear regression y = Xb

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www.soes.soton.ac.uk/teaching/courses/oa311/tides_3.ppt

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Note: Tidal Harmonics can cause tidal cycle to appear asymmetric.

Example: Harmonic analysis (cont’d) Southampton Surface Currents:Harmonic analysis for M2, M4=2xM2, M6=3xM2 ...


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Generalized linear models (GLM) are a flexible generalization of ordinary least squares regression. They relate the random distribution of the measured variable of the experiment (the distribution function) to the systematic (non-random) portion of the experiment (the linear predictor) through a function called the link function.

Generalized additive models (GAMs) are another extension to GLMs in which the linear predictor η is not restricted to be linear in the covariates X but is an additive function of the xi

s:

The smooth functions fi are estimated from the data. In general this requires a large number of data points and is computationally intensive.


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Data Handling MatlabUseful Tidbits …

Useful Tidbits

• regress - performs multiple linear regression using least squares

• nlinfit - performs nonlinear least-squares regression.

• glmfit - fits a generalized linear model.

sundermeyer mar 999 spring 2009 1 laboratory in oceanography: data and methods mar599, spring 2009...

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