Results_sst_olr_Elizabeth

Elizabeth Wong
 * SST and OLR **

**2. Field1 means and variances: space, time, spacetime ** 1. The GRAND (space+time) **mean** of my x and y: **27.5744 degC and 239.9729 (W/m^2). ** 2. The GRAND (space+time) **variance** of my x and y: ** 2.9326 (degC)^2 and 551.1539 (W/m^2)^2. ** 3. The GRAND standard deviations are: 1.7125 degC, 23.4770 W/m^2 4. The SPATIAL variance of my TIME MEAN longitude section is: **1.8427 (degC)^2 and 333.9493 (W/m^2)^2 ** 5. The TEMPORAL variance of my LONGITUDE MEAN time series is: **0.3616 (degC)^2 and 23.7800 (W/m2)^2 **





**3. (the assignment part) ** <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">1. Confirm that the time mean of the anomalies as defined above is 0. <span style="color: #008000; font-family: 'Arial Black',Gadget,sans-serif; font-size: 12pt;">Yes <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">2. Is the spatial mean of the anomalies (as defined above) 0?<span style="color: #008000; font-family: 'Arial Black',Gadget,sans-serif; font-size: 12pt;"> No <span style="color: black; font-family: Wingdings; font-size: 10pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">§ o <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt;">Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing? <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">3. My CLIMATOLOGICAL ANNUAL CYCLE have variance: o <span style="color: #008000; font-family: Arial,sans-serif; font-size: 12pt;">

**<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">var( climx12,1) 2.4852 **<span style="color: #008000; font-family: Arial,sans-serif; font-size: 12pt;"> (sst) **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">var( climy12,1) 410.5632 (olr) ** <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">4. My INTERANNUAL ANOMALY ARRAYS have variance: o **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">var( anomx,1) 0.4473 ** **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">var( anomy,1) 140.5907 ** <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts. **__<span style="color: black; font-family: Arial,sans-serif; font-size: 12pt;">6. Discuss your results __****<span style="color: black; font-family: Arial,sans-serif; font-size: 12pt;">: **
 * || ====<span style="font-family: 'Times New Roman',serif;">SST ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">a) total variance of x ==== || ==== **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">2.9326 ** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">b) purely spatial (variance of TIME mean at each lon) ==== || ==== **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">1.8427 ** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">c) variance of (x minus its TIME mean at each lon) ==== || ==== **1.0898** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">d) purely temporal (variance of LON mean at each time) ==== || ==== **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.3616 ** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">e) variance of (x minus its LON mean at each time) ==== || ==== **2.5710** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">f) remove both means (space-time variability) ==== || ====<span style="color: #008000; font-family: Arial,sans-serif;">**0.7282** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">g) mean seasonal cycle ==== || ==== **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">2.4852 ** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">h) deseasonalized anomalies ==== || ==== **<span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.4473 ** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">i) variance of longitudinal mean of h ==== || ==== **0.1025** ==== ||
 * ====<span style="font-family: 'Times New Roman',serif; font-size: 12pt; line-height: normal; margin: 9.35pt 0in;">j) h minus i ==== || ==== **0.3448** ==== ||

· <span style="color: #008000; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">(a) = (g) + (h). Yes it checks out.

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**<span style="color: black; font-family: Arial,sans-serif; font-size: 15.5pt;">4. Further decomposition of anomx by scale (using rebinning). **  o <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt;">What space and time scales (units: degrees and months) have the most variance in your anomx field?

§ <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">Matlab prints it: § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">variance_by_scalefactor = <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;">0.4473 0.4428 0.4333 0.4167 0.3826 0.1663 § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.4221 0.4187 0.4112 0.3976 0.3683 0.1604 § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.3860 0.3835 0.3780 0.3672 0.3431 0.1497 § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.3265 0.3247 0.3208 0.3130 0.2949 0.1305 § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.1874 0.1864 0.1841 0.1796 0.1697 0.0755 § <span style="color: green; font-family: Arial,sans-serif; font-size: 12pt;">0.0504 0.0500 0.0493 0.0475 0.0447 0.0178

<span style="color: black; font-family: Arial,sans-serif; font-size: 17pt; line-height: 17.75pt; margin-left: 0in; text-indent: -0.25in;"> o <span style="color: black; font-family: Arial,sans-serif; font-size: 12pt;">Create a contour plot of anomx -- can you see these "characteristic" scales by eye? Annotate your anomx plot with some ovals of about the right size (in powerpoint may be easiest), and put a few of these ovals in regions where you think you can see structures of about the right scales. I am looking for good eyeball judgement here.

<span style="color: black; font-family: Arial,sans-serif; font-size: 12pt; line-height: 0px; margin-left: 0in; overflow: hidden; text-indent: -0.25in;"> <span style="color: black; font-family: Arial,sans-serif; font-size: 17pt; line-height: 17.75pt; margin-bottom: 0in;">

==<span style="font-size: 1.3em; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 5px;">**5. Scatter plot, correlation and covariance, regression-explained variance** ==
 * 1) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">Based on your data fields (which you've seen pictures of), **make subsets of your 2 variables** x and y and **make a scatter plot of these showing the strongest (positive or negative) correlation of one field with the other you can find**. The subset might simply be all (x,t) values if your fields are very similar (olr, precip), or maybe the 240 time values at one longitude, or 144 longitudinal values in the time mean, or time series at different longitudes if some variability is offset in your two fields (like pressure and wind).
 * 2) <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">Do a scatter plot at longitude 225
 * 3) [[image:scatterhist_xy_225.jpg width="448" height="336"]][[image:scatterhist_climxy_225.jpg width="448" height="336"]]
 * 4) [[image:scatterhist_225.jpg]]


 * 1) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">Now consider the covariance and correlation of the two subset arrays entering your scatterplot.
 * <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">What is the correlation coefficient corresponding to this scatter plot?
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"> 1.0000 -0.7588
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"> -0.7588 1.0000
 * 1) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">What are the standard deviations of your two data subsets? <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">std(x)=1.2902 stdev(y)=15.9138
 * 2) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">What fraction of the variance of y can be 'explained' by linear regression on x (y = mx + b)? How does this relate to rho? How much y variance is explained? (variance: with units of y squared) What is m? //Hint: these are simple questions: use the math formula, not a computer code (Hsieh section 1.4.2, Eq. 1.33).//
 * <span style="color: #008000; display: inline !important; margin-bottom: 0in; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">The fraction of the variance of sst can be 'explained' by linear regression on olr is rho.^2=0.5758.
 * <span style="color: #008000; display: inline !important; margin-bottom: 0in; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">The fraction of the variance of sst can be 'explained' by linear regression on olr is rho.^2=0.5758.
 * 1) <span style="font-style: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">
 * <span style="color: #008000; margin-bottom: 0in; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">The explained variance of sst is std(x)^2*rh o.^2=0.9584
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">m is the slope of the least-squares fit line rho*std(y)/std(x)=9.359
 * 1) <span style="color: #ff0000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">What fraction of the variance of x can be 'explained' by linear regression on variable y? (x = nx + a)? How does this relate to rho? What is n? //Hint: these are simple questions, use the math formula not computer code.//
 * 2) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">Now add uncorrelated (random) noise with variance 1 to one of your variables. This might be like observation error. (In this case, it was added to OLR)
 * 3) How did the variance of y change when this noise was added?
 * 4) var(y)=551.1699, var(noisey)=552.4670
 * 5) Tried doing it for SST variable also, we get var(x)=2.9326, var(noisex)=3.9302
 * 6) My original OLR variance @ 225 ° was 253.249. The variance of my sst + random noise is: 255.2928. The addition of noise increases the variance.
 * 7) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">
 * <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">How did the correlation change? <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">rho = corrcoef(x,noisey)
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"> 1.0000 -0.4975
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"> -0.4975 1.0000
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">@ 255 degrees lon:
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">
 * 1.0000 -0.7607
 * -0.7607 1.0000
 * 1) <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">How do these changes affect the regression of y on x? How much (y+noise) variance is explained by linear regression on x? What is the new value of m in the new (noisey = mx + b) regression?
 * <span style="margin: 0.5em 0px 0px; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">Hint: all these could be answered without using the computer, but it may help to confirm with data
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">std(noisey)=15.9138
 * <span style="color: #008000; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;">rho^2 = 0.7607^2 = 0.5786
 * slope m, rho*std(y)/std(x)=9.382

> >

> ==<span style="font-size: 1.3em; margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 5px;">**6. Lagged correlation, covariance, and cross-covariance: hey let's compute all vs. all** == > Show the zero-lag spatial covariance and correlation structures for your primary field, like [|OLR_anoms_covar_correl.BEM.Matlab.png] this for OLR. (please label the axes better than I did!) Interpret the results. > > > <span style="color: black; font-family: Arial,sans-serif; font-size: 17pt; line-height: 23px;"> > > > just did the OLR for fun > > largest variance seen in SST is from 150-300 longitude (need to take x axis and multiply by 2.5 because it was scaled to 0-144). > For OLR, the largest variance seen is from 150-250 longitude. > For the zero-lag correlation plots, SST and OLR have similar graphs with a negative correlation from around 150-250 longitude which may indicate to us el nino/la nina events happening > > Show longitude-lag sections of the covariance or correlation of this field, for a base point at some longitude of interest. Like this for OLR at a central Pacific longitude: [|OLR.lagregression.BEM.jpg] (Please label the axes better than I did in this example! I hate Matlab). Better in IDL: [|olr_lag_covariances.gif] > > Intepret the results in terms of the characteristic space and time scales of your anomalies. Can you see these characteristic scales in your original raw data like in olr_lag_covariances.gif? > > > Share a longitude-lag slice of your lagged co-variance matrix for your TWO fields. Label it, interpret it. > > > > >