statistics lectures1. Why? We want to find relationships between variables (spatially, temporally).2. Basic definitions  probability concepts, normal distribution3. Hypothesis testing  examples, types of statistics, pitfalls4. Regression linear least squares, correlation, multivariate regression5. Introduction to matrix methods EOF/PC analysisReference: Hartmann's Objective Analysis course notesATMS 552 (winter quarter)http://www.atmos.washington.edu/~dennis/DATADISTRIBUTIONSDATADISTRIBUTIONSDATADISTRIBUTIONSDATADISTRIBUTIONSDATADISTRIBUTIONSDATADISTRIBUTIONSrelationshipscan we find relationships between variables?magnitudeHow related are the variables? How different are variables? How well can we predict one from another? Is Seattle colder in December than it is in June?http://www.ncdc.noaa.gov/oa/ncdc.html (SEATAC temperature in C)-10-505101520sample mean 2.3CDec 1999, 5 dayssample mean 16CJun 1999, 5 daysrelationshipscan we find relationships between variables?magnitudeHow related are the variables? How different are variables? How well can we predict one from another? Is Seattle colder in December than it is in June?http://www.ncdc.noaa.gov/oa/ncdc.html (SEATAC temperature in C)-10-505101520-10-505101520sample mean 2.3CDec 1999, 5 dayssample mean 16CJun 1999, 5 dayssample mean 3.1CDec 1997-1999, all days Jun 1997-1999, all dayssample mean 17Crelationshipscan we find relationships between variables?magnitudeHow related are the variables? How different are variables? How well can we predict one from another? Is December rainier in Seattle or Vancouver?http://www.ncdc.noaa.gov/oa/ncdc.html (SEATAC and Vancouver precip)sample mean 0.81cm/dSEA 5 days in Dec 1999sample mean 1.3cm/dYVR 5 days in Dec 19991234relationshipscan we find relationships between variables?magnitudeHow related are the variables? How different are variables? How well can we predict one from another?reliabilityHow representative is the result of the "truth"? Is December rainier in Seattle or Vancouver?http://www.ncdc.noaa.gov/oa/ncdc.html (SEATAC and Vancouver precip)sample mean 0.81cm/dSEA 5 days in Dec 1999sample mean 1.3cm/dYVR 5 days in Dec 1999sample mean 0.57cm/dSEA Dec 1994-199912341234YVR Dec 1994-1999sample mean 0.67cm/dbasic definitionsprobability density function:  moments of the distribution: r=1mean (about zero)   r=2variance r=3skewness r=4kurtosisprobability: subject to This is probability that a random  realisation of x will fall between  the limits [a,b]. f(x)basic definitionsthe normal distribution0 2 4-4 -2probability density function:moments of the distribution: r=1mean   m1=0 (about zero) r=2variance m2=1 r=3skewness m3=0 r=4kurtosis m4=3probability: subject to~~~+++basic definitionsthe normal distribution0 2 4-4 -2probability density function:moments of the distribution: r=1mean  m1=0 (about zero) r=2variance m2=1 r=3skewness m3=0 r=4kurtosis m4=3probability: subject to~~~+++00.20.40.60.8101basic definitionsthe normal distribution0 2 4-4 -2-4 -2 0 2 4probability density function:moments of the distribution: r=1mean  m1=0 (about zero) r=2variance m2=1 r=3skewness m3=0 r=4kurtosis m4=3probability: subject tocumulative distribution function:hypothesis testing1. State the desired significance level.2.State the null hypothesis H0 and  its alternative H1.3.Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the  conclusion.1. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the conclusion.hypothesis testingexamplesIs the mean of two samples different?(What is the confidence interval for the meanof these measurements?)H0: The means of these precip datasets are equal.H1: The means of these precip datasets are not equal. 5 10 15 20 25 305 10 15 20 25 30sample mean 0.57cm/dSEA Dec 1994-1999 YVR Dec 1994-1999sample mean 0.67cm/dday1234(cm)day1. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the conclusion.hypothesis testingexamplesIs the variance of two samples different?Will global warming lead to a climate with more heatwaves, cold snaps, hurricanes, floods, droughts... ?http://eces.org/articles/000741.phpSchaer et al. Nature 2004JJA temperature anomaly (C) compared to 1961-1990 mean1. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the conclusion.hypothesis testingexamplesIs the variance of two samples different?Will global warming lead to a climate with more heatwaves, cold snaps, hurricanes, floods, droughts... ?The 2003 heatwave in Europe was a1-in-46,000 event that claimed over 20,000 lives and cost £7 billiion.http://eces.org/articles/000741.phpSchaer et al. Nature 2004JJA temperature anomaly (C) compared to 1961-1990 meanSwiss JJA temperature distribution for 1864-2003temperature (C)frequency2003σ = 0.94CT'/σ = 5.41. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the conclusion.hypothesis testingexamplesIs there some non-zero correlation betweentwo variables?seasonal correlation of DJFM Nino3 withDJFM 1950-2000 1000mb air temperatureNCEP/NCAR reanalysis NOAA-CIRES/CDC1. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a statistic.4. State the critical region.5. Evaluate the statistic and state the conclusion.hypothesis testingexamplesIs there some non-zero correlation betweentwo variables?paleoclimate variability in the tropical Atlantic (Cariaco Basin sediment reflectance) with the North Atlantic (δ18O from GISP2 ice core) Peterson et al. 2002 ScienceCariaco BasinGISP2 (Greenland)1910 1920 1930 19401. State the desired significance level.2. State the null hypothesis H0 and its alternative H1.3. Choose a