National Oceanic and
Atmospheric Administration
United States Department of Commerce

Correlations and Variances

Correlations and Variances

Question:

I need to compute correlations. How do I do that?

The script variance.jnl, included in the Ferret distribution, defines variables variance, correlation, and covariancebased on the input variables. The variances are computed for each time seriesso if the input variables are defined in, say, X, Y, and time then the varianceand correlations are functions of X and Y representing the correlation for eachtime series.

The variance script offers the following coaching lines about how torun it:

yes? go variance
... Variance and Covariance: Instructions:
Use the LET/QUIET command to define the variable(s) P (and Q) as
your variable(s) of interest (e.g. yes? LET/QUIET P = u[x=180,y=0])
The variance of P will be variable P_VAR (Q --> Q_VAR)
The covariance will be COVAR The correlation will be CORREL.
Type GO VAR_N to obtain n/n+1 statistical correction factor
...

Define variables P and Q as the inputs to the script (or to geta variance of only one variable, define P.) Both variables should have the same time axis. If they are on different grids, use a regridding transformation to regrid one variable in the time dimension to the grid of the other.

 yes? SET DATA coads_climatology
 yes? LET p = sst[X=180,Y=10]
 yes? LET q = airt[X=180,Y=10]
 yes? GO variance

 

Here are our definitions of P and Q, and some of the variables that the script defines:

 yes? SHOW VAR
 Created by DEFINE VARIABLE:
 >>> Definitions that replace any file variable of same name:
 P = SST[X=180,Y=10]
 Q = AIRT[X=180,Y=10]
 ...
 P_VAR = P_DSQ[L=@AVE]
 "VARIANCE OF P"
 Q_VAR = Q_DSQ[L=@AVE]
 "VARIANCE OF Q"

 P_VAR_MASK = P_DSQ_MASK[L=@AVE]
 "VARIANCE OF P WHEN Q PRESENT"
 Q_VAR_MASK = Q_DSQ_MASK[L=@AVE]
 "VARIANCE OF Q WHEN P PRESENT"

 COVAR = PQ_DSQ[L=@AVE]
 "COVARIANCE OF P AND Q"
 CORREL = COVAR / (P_VAR_MASK*Q_VAR_MASK)^.5
 "CORRELATION OF P AND Q"

Listing the variances, correlation, and covariance,

 yes? list p_var, q_var
 DATA SET: /home/ja9/tmap/fer_dsets/data/coads_climatology.cdf
 LONGITUDE: 179E
 LATITUDE: 9N
 TIME: 01-JAN 00:45 to 31-DEC 06:34
 Column 1: P_VAR is VARIANCE OF P
 Column 2: Q_VAR is VARIANCE OF Q
 P_VAR Q_VAR 
 I / *: 0.2455 0.1085
 yes? LIST correl, covar
 DATA SET: /home/ja9/tmap/fer_dsets/data/coads_climatology.cdf
 LONGITUDE: 179E
 LATITUDE: 9N
 TIME: 01-JAN 00:45 to 31-DEC 06:34
 Column 1: CORREL is CORRELATION OF P AND Q
 Column 2: COVAR is COVARIANCE OF P AND Q
 CORREL COVAR 
 I / *: 0.6347 0.1036

The comments in variance.jnl suggest running var_n.jnl to make the n/n+1 correction. It is to be run after variance.jnl, and it redefines the variances to make that correction. So correlation and covariance are also redefined, as correl and covar are defined in terms of p_var and q_var.

Show the definitions of p_var and correl; see how p_var has a new definitionafter running the script var_n.jnl

 yes? SHOW VAR p_var
 P_VAR = P_DSQ[L=@AVE]
 "VARIANCE OF P"

 yes? SHOW VAR correl
 CORREL = COVAR / (P_VAR_MASK*Q_VAR_MASK)^.5
 "CORRELATION OF P AND Q"

 yes? GO var_n

 yes? SHOW VAR p_var
 P_VAR = P_DSQ[L=@AVE] * NDNM1
 "VARIANCE OF P"


Note how correl has changed:

 yes? LIST correl
 VARIABLE : CORRELATION OF P AND Q
 FILENAME : coads_climatology.cdf
 FILEPATH : /home/ja9/tmap/fer_dsets/data/
 LONGITUDE: 179E
 LATITUDE : 9N
 TIME : 01-JAN 00:45 to 31-DEC 06:34
 0.6347

Now, if we want to define P and Q to be variables in X, Y, and time, we can see how the correlation varies in space, with higher correlations between sea and air temperature at latitudes where there is a stronger seasonal signal.

 

 yes? SET DATA coads_climatology
 yes? LET p = sst[x=150:220,y=0:40]
 yes? LET q = airt[x=150:220,y=0:40]
 yes? go variance

 yes? go var_n

Note how correl is a function of X and Y

 yes? STAT correl
 
 CORRELATION OF P AND Q
 LONGITUDE: 150E to 140W
 LATITUDE: 0 to 40N
 Z: N/A
 TIME: 01-JAN 00:45 to 31-DEC 06:34
 DATA SET: /home/ja9/tmap/fer_dsets/data/coads_climatology.cdf
 
 Total # of data points: 700 (35*20*1*1)
 # flagged as bad data: 0
 Minimum value: -0.18597
 Maximum value: 0.9938
 Mean value: 0.87395 (unweighted average)
 Standard deviation: 0.20548
 
 yes? shade correl