Correlations and Variances

Question:

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

The script variance.jnl defines variables variance, correlation, and covariance based on the input variables. The variances are computed for each time series so if the input variables are defined in, say, X, Y, and time then the variance and correlations are functions of X and Y representing the correlation for each time series.

The variance script offeres the following coaching lines about how to run it:

... 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 get a variance of only one variable, define P.)

  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 definition after 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

Last modified: June 24, 2004