Laboratory in Oceanography: Data and Methods

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Laboratory in Oceanography Data and Methods
Intro to the Statistics Toolbox

• MAR599, Spring 2009
• Miles A. Sundermeyer

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Intro to Statistics Toolbox Statistics
Toolbox/Descriptive Statistics

• Measures of Central Tendency
• Geometric Mean
• Harmonic Mean

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Intro to Statistics Toolbox Statistics
Toolbox/Descriptive Statistics

• 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., order2
  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)

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Intro to Statistics Toolbox Statistics
Toolbox/Descriptive Statistics
Examples of Skewness Kurtosis
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Intro to Statistics Toolbox Statistics
Toolbox/Descriptive Statistics

• 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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Intro to Statistics Toolbox Statistics
Toolbox/Descriptive Statistics
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(t) 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 Toolbox Statistics
Toolbox/Statistical Visualization

• Probability Distribution Plots
• Normal Probability Plots
• gtgt x normrnd(10,1,25,1)
• gtgt normplot(x)

gtgt x exprnd(10,100,1) gtgt normplot(x)
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Intro to Statistics Toolbox Statistics
Toolbox/Statistical Visualization

• Probability Distribution Plots
• Quantile-Quantile Plots
• gtgt x poissrnd(10,50,1) y poissrnd(5,100,1)
• gtgt qqplot(x,y)

gtgt x normrnd(5,1,100,1) gtgt y
wblrnd(2,0.5,100,1) gtgt qqplot(x,y)
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Intro to Statistics Toolbox Statistics
Toolbox/Statistical Visualization

• Probability Distribution Plots
• Cumulative Distribution Plots
• gtgt y evrnd(0,3,100,1)
• gtgt cdfplot(y)
• gtgt hold on
• gtgt x -200.110
• gtgt f evcdf(x,0,3)
• gtgt plot(x,f,'m')
• gtgt legend('Empirical', ...
• 'Theoretical', ...
• 'Location','NW')

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Intro to Statistics Toolbox Statistics
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/tool
  box/stats/index.html?/access/helpdesk/help/toolbox
  /stats/http//www.mathworks.com/support/product/p
  roduct.html?productST

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Intro to Statistics Toolbox Statistics
Toolbox/Probability Distributions/Supported
Distributions
Supported distributions (contd)
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Intro to Statistics Toolbox Statistics
Toolbox/Probability Distributions/Supported
Distributions
Supported statistics
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Intro to Statistics Toolbox Statistics
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 gt null hypothesis value (right-tail
  test)
• H1 µlt 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 a is 0.05. If p-value lt a the
  test rejects the null hypothesis if p-value gt a,
  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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Intro to Statistics Toolbox Statistics
Toolbox/Hypothesis Tests

• 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 s of
  the underlying normal distribution. A z-test
  assumes that s is known a t-test does not. Thus
  t-test must determine s from the sample.

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Intro to Statistics Toolbox Statistics
Toolbox/Hypothesis Tests

• ztest
• The test requires s (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.

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Intro to Statistics Toolbox Statistics
Toolbox/Hypothesis Tests

• ttest
• Like z-test, except the t-test does not require s
  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
  population
• s 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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Intro to Statistics Toolbox Statistics
Toolbox/Hypothesis Tests

• 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.

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Intro to Statistics Toolbox Statistics
Toolbox/Hypothesis Tests
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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• ANOVA (ANalysis Of VAriance)
• ANOVA is like a t-test among multiple (typically
  gt2) 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.
• a.j is a matrix whose columns are the group
  means. (The "dot j" notation means a applies to
  all rows of column j. That is, aij is the same
  for all i.)
• eij 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.

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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• 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?
• gtgt load hogg
• gtgt 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
• gtgt p,tbl,stats anova1(hogg)
• gtgt 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.

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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• One-way ANOVA (contd)
• 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 eij 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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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• 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
• gtgt load hogg
• gtgt p,tbl,stats anova1(hogg)
• gtgt c,m multcompare(stats)
• Example
• see Light_DO.m

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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• 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).
• m is a constant matrix of the overall mean of
  the observations.
• a.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.
• b.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.
• gij is a matrix of interactions. Values in each
  row sum to 0, and values in each column sum to 0.
• eij 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.

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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance
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? gtgt load mileage mileage
33.3000 34.5000 37.4000 33.4000
34.8000 36.8000 32.9000 33.8000
37.6000 32.6000 33.4000 36.6000 32.5000
33.7000 37.0000 33.0000 33.9000 36.7000
gtgt cars 3 gtgt p,tbl,stats
anova2(mileage,cars)p,tbl,stats anova1(hogg)
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Intro to Statistics Toolbox Statistics
Toolbox/Analysis of Variance

• Two-way ANOVA (contd)
• 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 eij 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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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis

• 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 e. Given data on x1, x2, and y,
  regression estimates the model parameters ßj (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

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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis
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 1n.
This in turn can be written as y X b,
or
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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis
yi mxi b y X b
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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis

• Example Harmonic Analysis
• sin(qf) sin(q)cos(f) sin(f)cos(q)
• Let ACcos(f), BCsin(f)
• gt Csin(wtf) Asin(wt) Bcos(wt)
• Linear regression y Xb

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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis
Example Harmonic analysis (contd) Southampton
Surface Currents Harmonic analysis for M2,
M42xM2, M63xM2 ...

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

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Intro to Statistics Toolbox Statistics
Toolbox/Regression Analysis
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 ? is not restricted to be linear
in the covariates X but is an additive function
of the xis 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 Matlab Useful Tidbits

• Useful Tidbits
• regress - performs multiple linear regression
  using least squares
• nlinfit - performs nonlinear least-squares
  regression.
• glmfit - fits a generalized linear model.