Results_precip_u1000_David

HW3 - More Stats

Precipitation and Zonal Wind means & variances: space, time, spacetime

 * 1) The GRAND (spacetime) **mean**of my x and y (including units): 4.74 mm/d, -1.80 m/s
 * 2) The GRAND (spacetime) **variance**of my x and y (including units): 12.63 (mm/d)^2, 6.02 (m/s)^2
 * 3) The GRAND standard deviations are: 3.55 mm/d, 2.45  m/s
 * 4)  The SPATIAL variance of my TIME MEAN longitude section is: _(units). 5.76 (mm/d)^2, 4.54  (m/s)^2
 * 5)  The TEMPORAL variance of my LONGITUDE MEAN time series is: _(units). 094(mm/d)^2, 0.07  (m/s)^2

**3. (the assignment part)**

 * 1) Confirm that the 20-year mean of the anomalies as defined above is 0. Write math (on paper, for yourself) that proves it/ shows why.
 * 2) Is the spatial mean of the climate anomalies (as defined above) 0? Is it the same as the time series of the spatial mean of the raw data? Or is it a new object?
 * 3) My CLIMATOLOGICAL ANNUAL CYCLE has variance: _(units). 7.93 mm/d, 5.15 m/s
 * 4) My INTERANNUAL ANOMALY ARRAYS has variance: _(units). 4.69 mm/d, 6.02 m/s
 * 5) Fill out a variance decomposition table for field 1:
 * || Precip (mm/d) || Uwnd (m/s) ||
 * a) total variance of x || 12.63 || 6.02 ||
 * b) purely spatial (variance of TIME mean at each lon) || 5.76 || 4.54 ||
 * c) temporal anomalies (x minus its TIME mean at each lon) || 6.87 || 1.48 ||
 * d) purely temporal (variance of LON mean at each time) || 0.94 || 0.07 ||
 * e) spatial anomalies (x minus its LON mean at each time) || 11.69 || 5.95 ||
 * f) remove both means (space-time variability) || 12.63 || 6.02 ||
 * g) mean seasonal cycle || 7.93 || 5.16 ||
 * h) deseasonalized anomalies || 4.69 || 0.87 ||
 * i) variance of longitudinal mean of part h) || 0.25 || 0.04 ||
 * j) h minus i (anomalies from both space and monthly-climatological means) || 4.45 || 0.83 ||



**5. Scatter plot, correlation and covariance, regression-explained variance**

 * 1) Based on your data fields (which you've seenpictures 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).

Correlation between Precip / Uwind
 * x = precip**
 * y = Uwind**



Findings suggest a correlation with positive slope, and heaviest precip values near where Uwind = 0. Absolute values of Uwind are lower in the tropics than in the mid-lats, so this could be the ITCZ?


 * 1) Now consider the covariance and correlation of the two subset arrays entering your scatterplot.
 * What is the correlation coefficient corresponding to this scatter plot? 0.435
 * What are the standard deviations of your two data subsets? 3.55, 2.45
 * 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). 1-(.435)^2 = 0.811, so only 0.189 is explained //
 * 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.//


 * 1) 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))
 * How did the variance of y change when this noise was added? 6.0238, 7.0035 - increased
 * How did the correlation change? 0.4046 - decreased
 * 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? ANSWER: EXPLAINED VARIANCE IS UNCHANGED.

6.