Results_slp_precip_Amelia

**Sea Level Pressure and Precipitation means and variances: space, time, spacetime**

 * 1) The GRAND (spacetime)**mean**of my x and y (including units): 1010.97 hPa, 4.74mm
 * 2) The GRAND (spacetime)**variance**of my x and y (including units): 2.37 hPa^2, 12.63 mm^2
 * 3) The GRAND standard deviations are: 1.54 hPa, 3.55 mm
 * 4) The SPATIAL variance of my TIME MEAN longitude section is: _(units). 1.16 hPa^2, 5.76 mm^2
 * 5) The TEMPORAL variance of my LONGITUDE MEAN time series is: _(units). 0.55 hPa^2, 0.94 mm^2

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.
Yes, must be true by definition of anomaly 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? No CLIMATOLOGICAL ANNUAL CYCLE has variance: _(units). - 1.8683 hPa^2, 7.93 mm/s INTERANNUAL ANOMALY ARRAYS has variance: _(units). - 0.5037 hPa^2, 4.69 mm/s

Fill out a variance decomposition table for field 1:
 * || _SLP_ || _Precip_ ||
 * a) total variance of x || 2.37 || 12.63 ||
 * b) purely spatial (variance of TIME mean at each lon) || 1.16 || 5.76 ||
 * c) temporal anomalies (x minus its TIME mean at each lon) || 1.21 || 6.87 ||
 * d) purely temporal (variance of LON mean at each time) || 0.55 || 0.94 ||
 * e) spatial anomalies (x minus its LON mean at each time) || 1.82 || 11.69 ||
 * f) remove both means (space-time variability) || 0.66 || 5.93 ||
 * g) mean seasonal cycle || 1.87 || 7.93 ||
 * h) deseasonalized anomalies || 0.50 || 4.69 ||
 * i) variance of longitudinal mean of part h) || 0.16 || 0.25 ||
 * j) h minus i (anomalies from both space and monthly-climatological means) || 0.34 || 4.45 ||

**3. Further decomposition of spacetime variations: Mean seasonal cycle and 'climate anomaly'**
Mean Seasonal Cycle for SLP: Climate Anomaly for SLP: Mean Seasonal Cycle for Precip: Climate Anomaly for Precip:

**4. Further decomposition of anomx by scale (using rebinning).**


= **Assignment for Homework 3: Parts 5-6: characterizing co-variability between 2 fields.** =

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



What is the correlation coefficient corresponding to this scatter plot? -0.46

What are the standard deviations of your two data subsets? 1.54, 3.55

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?

not explained = 1 - rho^2 = 0.79, so only 21% of the variance is accounted for by the regression.

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?


 * Now add uncorrelated (random) noise with variance 1 to one of your variables.**

How did the variance of y change when this noise was added? 12.63 to 13.60 - Increase

How did the correlation change? -0.44 - Decrease

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.