Result_uwnd_olr_Hosmay

=Uwnd and OLR=

Hosmay Lopez

2. Field1 means and variances: space, time, spacetime

 * 1) The GRAND (space+time) **mean** of my x and y: ** -1.7873 m/s and 240.056 (W/m^2). **
 * 2) The GRAND (space+time) **variance** of my x and y: ** 6.0044 (m/s)^2 and 549.196 (W/m^2)^2. **
 * 3) The GRAND standard deviations are: 2.4504m/s, 23.4349 W/m^2
 * 4) The SPATIAL variance of my TIME MEAN longitude section is: ** 4.4327 (m/s)^2 and 339.902 (W/m^2)^2 **
 * 5) The TEMPORAL variance of my LONGITUDE MEAN time series is: ** 0.07107 (m/s)^2 and 23.78 (W/m^2)^2 **

3. (the assignment part)

 * 1) Confirm that the time mean of the anomalies as defined above is 0. ** Yes **
 * 2) Is the spatial mean of the anomalies (as defined above) 0?
 * ** No **
 * Is it the same as the time of the spatial mean of the raw data? Or is it a new thing?
 * ** Removing the seasonal cycle from the raw data (spatially averaged) should give you similar results as taking the spatial average of interannual anomalies. This is shown in the nest figures (bottom panel). **



>> ** var( climy12,1) 410.431 (W/m^2)^2 (OLR) ** >> ** var( anomy,1) 138.763 (W/m^2)^2 (OLR) **
 * 1) My CLIMATOLOGICAL ANNUAL CYCLE have variance:
 * ** var( climx12,1) 5.1489 (m/s)^2 (uwnd) **
 * 1) My INTERANNUAL ANOMALY ARRAYS have variance:
 * ** var( anomx,1) 0.8551 (m/s)^2 (uwnd) **

5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.
 * [[image:uwnd_season.gif width="520" height="371"]] || [[image:olr_season.gif width="489" height="367"]] ||
 * [[image:uwnd_anom.gif width="520" height="328"]] || [[image:olr_anom.gif width="490" height="325"]] ||

**__6. Discuss your results__:** Write down all the decompositions of total variance that add up sensibly, and confirm them by noticing that the numbers work out. Example: (a) = (g) + (h). Yes it checks out, 100 = 23 + 77. or whatever numbers for your field x
 * || uwnd || OLR ||
 * a) total variance of x || 6.0044 || 549.196 ||
 * b) purely spatial (variance of TIME mean at each lon) || 4.5285 || 332.64 ||
 * c) temporal anomalies (x minus its TIME mean at each lon) || 1.4759 || 216.556 ||
 * d) purely temporal (variance of LON mean at each time) || 0.07107 || 23.78 ||
 * e) spatial anomalies (x minus its LON mean at each time) || 5.9332 || 525.417 ||
 * f) remove both means (space-time variability) || 1.4048 || 192.776 ||
 * g) mean seasonal cycle || 5.1489 || 410.431 ||
 * h) deseasonalized anomalies || 0.8551 || 138.763 ||
 * i) variance of longitudinal mean of part h) || 0.3475 || 6.478 ||
 * j) h minus i (anomalies from both space and monthly-climatological means) || 0.8354 || 134.113 ||

(a) = (g) + (h) (a) = (b) + (c) (a) = (b) + (d) + (f) (a) = (c) + (e) - (f) (a) = (d) + (e)

**Further decomposition of anomx by scale (using rebinning).**
What space and time scales (units: degrees and months) have the most variance in your anomx field?

Larger gradient of variance by scale factor at large scales and short time scales for both OLR and Uwnd. The Variance by scale factor indicates that variance decreases faster for different time scales than for spatial scales variance_by_scalefactor = 0.8684 0.8585 0.8287 0.7558 0.6191 0.3109 0.7036 0.6965 0.6751 0.6214 0.5152 0.2628  0.5948 0.5897 0.5737 0.5315 0.4489 0.2339  0.4551 0.4514 0.4401 0.4100 0.3514 0.1872  0.2962 0.2937 0.2861 0.2657 0.2271 0.1302  0.1325 0.1312 0.1273 0.1180 0.1021 0.0577

5. Scatter plot, correlation and covariance, regression-explained variance
First, I found the largest correlation between Uwnd and OLR to be at 235E in longitude. The value is negative: -0.756. So, I based my time series for the scatterplots on this location. No surprisingly, this is about the longitude where SST anomalies are maximum during ENSO events. Atmospheric deep convection are located just west of this longitude, out of phase with the SST anomalies.
 * 1) 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).
 * [[image:scat_uwnd_olr.gif width="445" height="387"]] || [[image:scat_olr_uwnd.gif width="409" height="387"]] ||

2. Now consider the covariance and correlation of the two subset arrays entering your scatterplot.
 * What is the correlation coefficient corresponding to this scatter plot? rho = corrcoef(x,y) = - 0.756
 * What are the standard deviations of your two data subsets? std(x) = 1.1623 ; std(y) = 13.7632
 * 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).//

From section 1.4.2 (relating regression to correlation) of the class handout, we see that the explained variance of y by linear regressing on x is equal to the correlation coefficient squared. That is (-0.757)^2 = 0.573
 * 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.//

It should be the same as that for Y because they have the same correlation coefficient. The difference is just in the variance explained, not the fractions of variances. So, the variance explained fraction = 0.573

3. Now add uncorrelated (random) noise with variance 1 to one of your variables. This might be like observation error. noisey = y + random('Normal',0,1,size(y)) Here, I added the noise in the OLR variable!!! The slope of the regression (i.e. m) remain unchanged, due to similar correlations. The constant b (or y-intercept) changed with the addition of random noise. The fraction of variance explained is also unchanged due to similar correlations. So, variance explained = 0.573.
 * How did the variance of y change when this noise was added? var(y) = 189.39 ; var(noisey) = 190.45 [W/m^2]^2 . It did not change much.
 * How did the correlation change? rho = corrcoef(x,noisey) = - 0.757. The correlation stayed the same because the random noise is uncorrelated with the 2 variables, so it will NOT have an effect on the correlation of X and Y.
 * 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?


 * [[image:scat_uwnd_olr-noise.gif width="396" height="302"]] || [[image:scat_olr-noise_uwnd.gif width="420" height="301"]] ||

6. Lagged correlation, covariance, and cross-covariance: questions

 * 1) 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.


 * [[image:q61.jpg width="622" height="446"]] ||
 * [[image:q61c.jpg width="697" height="348"]] ||
 * [[image:q61b.jpg width="611" height="388"]] ||


 * 1) 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

Cross-sections were taken at 80E, 140E, 180E, and 330E.

We can compare the characteristic space and time scales in the U-wind anomaly plot (Left plot) and those of the 4 panel plot in the right. The characteristic scales match with each other at a given cross section. 
 * 1) 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?
 * 1) Share a longitude-lag slice of your lagged co-variance matrix for your TWO fields. Label it, interpret it.

The figure shows the lag-lead covariance matrix of the OLR time-series at longitude 235E (or 125W) with U-wind. Note that the highest covariance is not co-located with the longitude at which OLR time series was taken (i.e. 125W), it is further west (at about 180). Also, the maximum covariance occurs at Lag = + 4 months. This suggest that OLR anomalies at about 125W (central Pacific) are related to zonal wind anomalies further west an about 4 months earlier. This is probably related to equatorial Kelvin waves, which we can detect by the eastward propagation (from top-left to bottom-right) of the covariance signal. It is interesting to see relative high covariance at high lags (i.e 20 months). This is definitively the ENSO signal, since this oscillation pattern has a period of about 40 months and do not appear to be totally damped.