10
$\begingroup$

In the following example I have a data frame which consists of a time series of water temperature measurements recorded at 5 depths in the ocean where each value in Temp corresponds to the date in DateTime and the depth in Depth.

set.seed(1)
Temp <- rnorm(43800,sd=20)
AirT <- rnorm(8760,sd=20)
Depth <- c(1:5)

DateTime = seq(from=as.POSIXct("2010-01-01 00:00"), 
           to=as.POSIXct("2010-12-31 23:00"), length=8760)
Time <- as.POSIXct(DateTime, format = "%Y-%m-%d %H:%M")

DatT <- data.frame(Temp)

## bind together
FinalDat <- cbind(DatT, Date = rep(Time,5))

FinalDat <- cbind(FinalDat, AirT = rep(AirT, 5),
                  Depth = rep(Depth, each = 8760),
                  Doy = as.numeric(format(Time,format = "%j")),
                  DecTime = rep(seq(1,366, length = 8760),5))

From this I would like to find the similarity between air temperature (AirT) and the water temperature at these different depths. I can calculate the correlation between air temperature and water temperature for the different depths which shows that the correlation decreases with depth, however, I am looking for a more sophisticated method (other than receiving a single value for each time series). Can anyone suggest an appropriate method for achieving this?

Here is a thinned version of my data.frame (FinalDat):

       Temp       Date     AirT Depth
1    3.4069 2011-01-01  6.52710   0.5
2    3.2165 2011-01-08  6.77050   0.5
3    3.5630 2011-01-15  6.61410   0.5
4    3.8027 2011-01-22 -2.94440   0.5
5    3.7328 2011-01-29 -2.92900   0.5
6    4.1008 2011-02-05 12.74800   0.5
7    5.1825 2011-02-12  8.91900   0.5
8    5.0460 2011-02-19  4.43910   0.5
9    5.6156 2011-02-26  8.58090   0.5
10   5.5237 2011-03-05  1.56630   0.5
11   5.5110 2011-03-12  7.94470   0.5
12   6.5645 2011-03-19 -0.25485   0.5
13   7.8872 2011-03-26  4.11280   0.5
14   8.9096 2011-04-02 12.88700   0.5
15  10.8740 2011-04-09  6.34800   0.5
16  10.9600 2011-04-16  9.44190   0.5
17  14.6930 2011-04-23 12.37300   0.5
18  15.1330 2011-04-30 11.40200   0.5
19  14.2740 2011-05-07 14.48400   0.5
20  14.5670 2011-05-14  5.68590   0.5
21  14.0330 2011-05-21  8.36570   0.5
22  13.3870 2011-05-28 11.03900   0.5
23  16.5560 2011-06-04 13.10300   0.5
24  16.4140 2011-06-11  7.56720   0.5
25  16.2590 2011-06-18 12.02400   0.5
26  16.5120 2011-06-25 11.43100   0.5
27  17.9290 2011-07-02  8.75940   0.5
28  18.7150 2011-07-09 13.06100   0.5
29  19.4970 2011-07-16 14.96000   0.5
30  17.7090 2011-07-23 10.06100   0.5
31  19.5090 2011-07-30 11.76200   0.5
32  20.4780 2011-08-06 12.56800   0.5
33  17.8850 2011-08-13 16.93200   0.5
34  17.7440 2011-08-20 14.82700   0.5
35  18.1400 2011-08-27 12.14800   0.5
36  17.0460 2011-09-03 17.19700   0.5
37  16.5960 2011-09-10 18.45800   0.5
38  15.6430 2011-09-17 13.07600   0.5
39  14.9960 2011-09-24 13.19300   0.5
40  16.9880 2011-10-01 20.17800   0.5
41  14.5380 2011-10-08 10.92600   0.5
42  14.3770 2011-10-15  9.64180   0.5
43  12.6210 2011-10-22 11.71200   0.5
44  11.7810 2011-10-29  7.32770   0.5
45  11.8270 2011-11-05 11.15600   0.5
46  10.9790 2011-11-12 11.30000   0.5
47  10.3410 2011-11-19 10.77100   0.5
48   9.6291 2011-11-26  6.81090   0.5
49   8.5466 2011-12-03  9.11540   0.5
50   7.1007 2011-12-10  3.52720   0.5
51   6.3081 2011-12-17  0.32245   0.5
52   6.1304 2011-12-24  5.40000   0.5
53   6.1974 2011-12-31 11.93100   0.5
54   3.2471 2011-01-07  0.44508   4.0
55   3.4851 2011-01-14  8.43040   4.0
56   3.8537 2011-01-21 -2.26530   4.0
57   3.8496 2011-01-28  1.33950   4.0
58   3.8461 2011-02-04 10.18700   4.0
59   4.7892 2011-02-11  6.91330   4.0
60   5.1160 2011-02-18  5.30300   4.0
61   5.4236 2011-02-25 10.40700   4.0
62   5.3190 2011-03-04  3.63990   4.0
63   5.5318 2011-03-11  5.49690   4.0
64   6.0956 2011-03-18  2.12060   4.0
65   6.7955 2011-03-25  5.66700   4.0
66   8.5056 2011-04-01 12.13400   4.0
67   9.8450 2011-04-08  6.11440   4.0
68  10.7220 2011-04-15  9.85800   4.0
69  11.3970 2011-04-22 11.39300   4.0
70  12.6660 2011-04-29  9.54510   4.0
71  13.6370 2011-05-06 12.99600   4.0
72  14.7530 2011-05-13  8.83800   4.0
73  13.8210 2011-05-20  8.81900   4.0
74  13.5120 2011-05-27 10.80500   4.0
75  14.2630 2011-06-03 11.08700   4.0
76  15.6330 2011-06-10  7.20970   4.0
77  15.5290 2011-06-17 10.11400   4.0
78  16.6340 2011-06-24 11.29400   4.0
79  17.4820 2011-07-01 11.43900   4.0
80  17.8620 2011-07-08 13.29700   4.0
81  18.8970 2011-07-15 11.61900   4.0
82  17.4720 2011-07-22 12.73800   4.0
83  17.8640 2011-07-29 14.95600   4.0
84  17.6610 2011-08-05 16.48600   4.0
85  17.4880 2011-08-12 16.06000   4.0
86  17.5980 2011-08-19 11.38900   4.0
87  17.8150 2011-08-26 10.69200   4.0
88  16.7470 2011-09-02 14.71100   4.0
89  16.0230 2011-09-09 14.09600   4.0
90  15.5350 2011-09-16 13.03500   4.0
91  15.0200 2011-09-23 11.81800   4.0
92  15.1200 2011-09-30 14.49800   4.0
93  15.0590 2011-10-07 10.76800   4.0
94  14.5790 2011-10-14 13.91800   4.0
95  12.7810 2011-10-21  8.84760   4.0
96  11.9060 2011-10-28  8.37600   4.0
97  11.7150 2011-11-04 12.57400   4.0
98  11.1170 2011-11-11 11.26200   4.0
99  10.3560 2011-11-18 11.11100   4.0
100  9.8484 2011-11-25 12.91700   4.0
101  8.7573 2011-12-02  2.43560   4.0
102  7.2890 2011-12-09  5.87070   4.0
103  6.4547 2011-12-16  2.31850   4.0
104  6.0336 2011-12-23 11.12900   4.0
105  6.2363 2011-12-30  4.78220   4.0
106  3.4241 2011-01-06  2.92880   6.0
107  3.4075 2011-01-13  9.59030   6.0
108  3.9650 2011-01-20 -1.45990   6.0
109  3.9061 2011-01-27  3.49050   6.0
110  3.8072 2011-02-03  6.15450   6.0
111  4.7597 2011-02-10  8.87270   6.0
112  4.9920 2011-02-17  4.61980   6.0
113  5.1485 2011-02-24  7.66730   6.0
114  5.4556 2011-03-03 -0.73976   6.0
115  5.4305 2011-03-10  9.08700   6.0
116  5.9204 2011-03-17  6.57230   6.0
117  6.4393 2011-03-24  4.21580   6.0
118  6.7361 2011-03-31  9.55530   6.0
119  8.9556 2011-04-07 10.24900   6.0
120 10.7240 2011-04-14  8.15550   6.0
121 10.4810 2011-04-21  9.75940   6.0
122 10.6310 2011-04-28  6.43770   6.0
123 13.2840 2011-05-05 10.70900   6.0
124 14.1380 2011-05-12  9.11810   6.0
125 13.7850 2011-05-19  8.26810   6.0
126 13.6800 2011-05-26 11.65000   6.0
127 13.8950 2011-06-02 13.10500   6.0
128 13.8530 2011-06-09 11.09500   6.0
129 14.6830 2011-06-16 12.63000   6.0
130 15.3570 2011-06-23 12.66100   6.0
131 15.5860 2011-06-30  9.57130   6.0
132 16.2190 2011-07-07 13.84300   6.0
133 16.1140 2011-07-14 10.93200   6.0
134 17.1240 2011-07-21 13.44100   6.0
135 17.2110 2011-07-28 13.41700   6.0
136 16.9800 2011-08-04 14.81500   6.0
137 17.4610 2011-08-11 16.33900   6.0
138 17.5690 2011-08-18 10.04300   6.0
139 17.3600 2011-08-25 11.87600   6.0
140 16.6180 2011-09-01 11.71000   6.0
141 16.1330 2011-09-08 11.84600   6.0
142 15.6770 2011-09-15 12.17800   6.0
143 15.1640 2011-09-22 11.90200   6.0
144 14.8160 2011-09-29 17.07200   6.0
145 15.0100 2011-10-06 13.21700   6.0
146 14.4410 2011-10-13 12.01300   6.0
147 13.0510 2011-10-20  3.75340   6.0
148 12.0580 2011-10-27  8.76700   6.0
149 11.6430 2011-11-03 13.96900   6.0
150 11.1400 2011-11-10 13.03200   6.0
151 10.5090 2011-11-17  8.26000   6.0
152  9.8690 2011-11-24 12.02100   6.0
153  8.9167 2011-12-01  9.64050   6.0
154  7.3953 2011-12-08  4.99180   6.0
155  6.5533 2011-12-15  3.73240   6.0
156  5.9886 2011-12-22 12.69300   6.0
157  6.3341 2011-12-29  7.12550   6.0
158  3.4249 2011-01-05  5.66070   8.0
159  3.4232 2011-01-12  4.42930   8.0
160  3.9632 2011-01-19  0.04384   8.0
161  3.9203 2011-01-26  5.33010   8.0
162  3.7569 2011-02-02  4.74030   8.0
163  4.6606 2011-02-09  6.48010   8.0
164  4.9632 2011-02-16  5.92770   8.0
165  4.9742 2011-02-23  7.15490   8.0
166  5.3302 2011-03-02  1.52250   8.0
167  5.3380 2011-03-09  6.46400   8.0
168  5.7341 2011-03-16  6.49500   8.0
169  6.1663 2011-03-23  4.01320   8.0
170  6.1420 2011-03-30 10.23300   8.0
171  7.1226 2011-04-06 12.10300   8.0
172  8.4905 2011-04-13  5.66120   8.0
173  8.7548 2011-04-20  7.94430   8.0
174  8.7160 2011-04-27  9.66080   8.0
175 10.1360 2011-05-04  5.13940   8.0
176  9.6597 2011-05-11 12.00800   8.0
177 12.8610 2011-05-18 13.42900   8.0
178 13.1300 2011-05-25  7.96200   8.0
179 13.6080 2011-06-01  8.94920   8.0
180 13.4880 2011-06-08 10.58700   8.0
181 13.6130 2011-06-15 14.82200   8.0
182 13.4320 2011-06-22 13.28300   8.0
183 13.3850 2011-06-29 10.09900   8.0
184 13.1230 2011-07-06 14.30500   8.0
185 13.2270 2011-07-13 13.44700   8.0
186 13.3370 2011-07-20 13.27900   8.0
187 13.4860 2011-07-27 16.17600   8.0
188 14.1970 2011-08-03 15.40500   8.0
189 14.0740 2011-08-10 11.20800   8.0
190 14.2160 2011-08-17 10.92000   8.0
191 14.7250 2011-08-24 13.57000   8.0
192 16.2060 2011-08-31 12.35700   8.0
193 15.6070 2011-09-07 13.77500   8.0
194 15.7620 2011-09-14 13.05900   8.0
195 15.1750 2011-09-21 11.50800   8.0
196 14.7600 2011-09-28 15.44600   8.0
197 14.7480 2011-10-05 15.41900   8.0
198 14.3420 2011-10-12 12.46600   8.0
199 13.3520 2011-10-19  8.04520   8.0
200 12.1960 2011-10-26  9.85760   8.0
201 11.6640 2011-11-02  9.94480   8.0
202 11.1650 2011-11-09  9.83490   8.0
203 10.6510 2011-11-16  7.18460   8.0
204 10.0600 2011-11-23  3.36370   8.0
205  9.1110 2011-11-30  6.19500   8.0
206  7.7550 2011-12-07  5.57610   8.0
207  6.7252 2011-12-14  5.46150   8.0
208  5.9302 2011-12-21  6.02540   8.0
209  6.4421 2011-12-28  8.87860   8.0
210  3.5953 2011-01-04 -0.87434  10.0
211  3.3527 2011-01-11  7.57030  10.0
212  3.9915 2011-01-18  4.65410  10.0
213  3.8234 2011-01-25  7.24100  10.0
214  3.8257 2011-02-01  2.03130  10.0
215  4.6403 2011-02-08  4.28250  10.0
216  5.0439 2011-02-15  2.09610  10.0
217  4.9152 2011-02-22  4.52120  10.0
218  5.3290 2011-03-01  4.09410  10.0
219  5.4156 2011-03-08 -0.97393  10.0
220  5.7129 2011-03-15  5.77950  10.0
221  6.0155 2011-03-22  4.84410  10.0
222  5.9742 2011-03-29  4.10630  10.0
223  6.1641 2011-04-05 10.80000  10.0
224  6.4420 2011-04-12  7.07030  10.0
225  6.6906 2011-04-19  9.72580  10.0
226  6.8168 2011-04-26 11.22400  10.0
227  6.5933 2011-05-03  7.08340  10.0
228  7.7907 2011-05-10 10.99000  10.0
229  6.8976 2011-05-17 12.02200  10.0
230  7.1110 2011-05-24  9.37830  10.0
231 10.1990 2011-05-31  7.85870  10.0
232  9.9413 2011-06-07  9.30420  10.0
233 10.2710 2011-06-14 10.07000  10.0
234  9.6727 2011-06-21 15.00500  10.0
235 10.0280 2011-06-28 13.74500  10.0
236 10.3870 2011-07-05 17.47700  10.0
237  9.9664 2011-07-12 15.32100  10.0
238  9.8633 2011-07-19 13.27000  10.0
239 10.1500 2011-07-26 11.90700  10.0
240  9.6085 2011-08-02 17.43300  10.0
241 10.2650 2011-08-09 13.26100  10.0
242 10.2090 2011-08-16 13.34400  10.0
243  9.9657 2011-08-23 16.62600  10.0
244 10.1730 2011-08-30 13.30700  10.0
245 10.4790 2011-09-06 14.72500  10.0
246 10.5400 2011-09-13 14.77800  10.0
247 11.9570 2011-09-20 15.70200  10.0
248 12.4710 2011-09-27 11.22700  10.0
249 12.1980 2011-10-04 13.29700  10.0
250 13.8790 2011-10-11 16.68600  10.0
251 13.7680 2011-10-18  7.88400  10.0
252 12.1430 2011-10-25 13.14200  10.0
253 11.5050 2011-11-01 13.24000  10.0
254 11.1610 2011-11-08 10.43400  10.0
255 10.7390 2011-11-15  9.17300  10.0
256 10.0850 2011-11-22  9.06990  10.0
257  9.1183 2011-11-29 12.74600  10.0
258  7.9380 2011-12-06  4.43310  10.0
259  6.8017 2011-12-13  9.05250  10.0
260  5.9501 2011-12-20  7.67070  10.0
261  6.3240 2011-12-27 10.47900  10.0
262  3.6034 2011-01-03  0.44518  12.0
263  3.2966 2011-01-10  1.92110  12.0
264  3.8193 2011-01-17  5.50270  12.0
265  3.8413 2011-01-24  2.38370  12.0
266  3.8157 2011-01-31 -0.20332  12.0
267  4.3896 2011-02-07 10.31200  12.0
268  4.9550 2011-02-14  6.72160  12.0
269  4.9230 2011-02-21  4.14600  12.0
270  5.3725 2011-02-28  4.17540  12.0
271  5.4386 2011-03-07 -0.17011  12.0
272  5.6663 2011-03-14 -0.20653  12.0
273  5.6173 2011-03-21  9.40060  12.0
274  5.7879 2011-03-28  3.42890  12.0
275  5.8027 2011-04-04  5.26820  12.0
276  5.7847 2011-04-11  8.31470  12.0
277  5.9425 2011-04-18  8.85620  12.0
278  6.2409 2011-04-25  6.88140  12.0
279  5.9921 2011-05-02  9.72310  12.0
280  6.0906 2011-05-09 12.04900  12.0
281  6.0394 2011-05-16 12.44600  12.0
282  6.0922 2011-05-23  9.88280  12.0
283  6.4765 2011-05-30 12.69800  12.0
284  6.6011 2011-06-06 11.33400  12.0
285  6.6846 2011-06-13 15.29900  12.0
286  6.9803 2011-06-20  9.87070  12.0
287  6.6561 2011-06-27 19.08300  12.0
288  7.2964 2011-07-04 12.62800  12.0
289  6.5017 2011-07-11 11.53500  12.0
290  7.4560 2011-07-18 13.64600  12.0
291  7.4972 2011-07-25  9.44130  12.0
292  7.1581 2011-08-01 18.45400  12.0
293  7.4089 2011-08-08 11.82800  12.0
294  7.5630 2011-08-15 12.00700  12.0
295  7.5153 2011-08-22 12.84500  12.0
296  7.4694 2011-08-29 11.80500  12.0
297  7.5982 2011-09-05 11.41900  12.0
298  7.5733 2011-09-12 13.39300  12.0
299  7.7114 2011-09-19 13.35600  12.0
300  7.9359 2011-09-26 15.57900  12.0
301  7.7604 2011-10-03 15.83000  12.0
302  8.2146 2011-10-10 16.05000  12.0
303  8.8018 2011-10-17 11.35400  12.0
304 12.0390 2011-10-24 14.38700  12.0
305 11.3230 2011-10-31 14.43600  12.0
306 11.4140 2011-11-07  2.19910  12.0
307 10.9020 2011-11-14  9.79220  12.0
308 10.1070 2011-11-21  9.87410  12.0
309  9.2805 2011-11-28  4.62830  12.0
310  8.2556 2011-12-05  2.57670  12.0
311  6.9548 2011-12-12  4.16250  12.0
312  6.0980 2011-12-19  1.10680  12.0
313  6.1540 2011-12-26 12.25600  12.0
314  3.6293 2011-01-02  2.14550  14.0
315  3.3969 2011-01-09  4.79400  14.0
316  3.8152 2011-01-16 11.32500  14.0
317  3.9532 2011-01-23  0.58015  14.0
318  3.8262 2011-01-30 -0.96998  14.0
319  4.1869 2011-02-06 10.84800  14.0
320  4.7450 2011-02-13  6.02670  14.0
321  4.9848 2011-02-20  1.09370  14.0
322  5.2429 2011-02-27  5.34090  14.0
323  5.3107 2011-03-06  4.29060  14.0
324  5.5456 2011-03-13  6.86140  14.0
325  5.6181 2011-03-20  6.71190  14.0
326  5.6648 2011-03-27  7.01350  14.0
327  5.6326 2011-04-03  7.26700  14.0
328  5.7413 2011-04-10  9.25000  14.0
329  5.7310 2011-04-17  7.36030  14.0
330  5.6897 2011-04-24  9.62540  14.0
331  5.8051 2011-05-01  9.68150  14.0
332  5.6844 2011-05-08 14.67800  14.0
333  5.7530 2011-05-15  9.46050  14.0
334  5.7696 2011-05-22 10.65700  14.0
335  5.8255 2011-05-29 12.23400  14.0
336  5.8422 2011-06-05 12.08500  14.0
337  5.9016 2011-06-12  5.82010  14.0
338  5.8791 2011-06-19 12.82400  14.0
339  5.9407 2011-06-26 15.17000  14.0
340  5.9502 2011-07-03 11.82400  14.0
341  5.8969 2011-07-10 10.83500  14.0
342  5.9582 2011-07-17 12.44000  14.0
343  6.1421 2011-07-24 12.59700  14.0
344  6.0970 2011-07-31 15.59200  14.0
345  6.1043 2011-08-07 12.65400  14.0
346  6.1677 2011-08-14 13.57600  14.0
347  6.3980 2011-08-21 15.59100  14.0
348  6.4571 2011-08-28 12.29000  14.0
349  6.0783 2011-09-04 15.58600  14.0
350  6.5784 2011-09-11 14.65100  14.0
351  6.5329 2011-09-18  8.72590  14.0
352  6.5395 2011-09-25 14.24400  14.0
353  6.8765 2011-10-02 14.41700  14.0
354  6.5151 2011-10-09 15.62400  14.0
355  6.5328 2011-10-16 10.04000  14.0
356  6.7127 2011-10-23 11.33300  14.0
357  7.0375 2011-10-30 11.93500  14.0
358  7.1903 2011-11-06  6.52240  14.0
359  8.4702 2011-11-13  7.77260  14.0
360  9.5075 2011-11-20  5.02910  14.0
361  9.3919 2011-11-27 13.07300  14.0
362  8.3426 2011-12-04  7.68420  14.0
363  6.9961 2011-12-11  5.19810  14.0
364  6.2100 2011-12-18  0.97878  14.0
365  6.0532 2011-12-25 10.49000  14.0
366  3.5866 2011-01-01  6.52710  16.0
367  3.6081 2011-01-08  6.77050  16.0
368  3.5889 2011-01-15  6.61410  16.0
369  3.9475 2011-01-22 -2.94440  16.0
370  3.8560 2011-01-29 -2.92900  16.0
371  4.0730 2011-02-05 12.74800  16.0
372  4.6510 2011-02-12  8.91900  16.0
373  5.0101 2011-02-19  4.43910  16.0
374  4.8702 2011-02-26  8.58090  16.0
375  5.2720 2011-03-05  1.56630  16.0
376  5.5106 2011-03-12  7.94470  16.0
377  5.6273 2011-03-19 -0.25485  16.0
378  5.6357 2011-03-26  4.11280  16.0
379  5.6058 2011-04-02 12.88700  16.0
380  5.6570 2011-04-09  6.34800  16.0
381  5.6360 2011-04-16  9.44190  16.0
382  5.6088 2011-04-23 12.37300  16.0
383  5.6481 2011-04-30 11.40200  16.0
384  5.6099 2011-05-07 14.48400  16.0
385  5.6895 2011-05-14  5.68590  16.0
386  5.6614 2011-05-21  8.36570  16.0
387  5.6061 2011-05-28 11.03900  16.0
388  5.6570 2011-06-04 13.10300  16.0
389  5.6595 2011-06-11  7.56720  16.0
390  5.6647 2011-06-18 12.02400  16.0
391  5.6749 2011-06-25 11.43100  16.0
392  5.6809 2011-07-02  8.75940  16.0
393  5.6509 2011-07-09 13.06100  16.0
394  5.6534 2011-07-16 14.96000  16.0
395  5.6758 2011-07-23 10.06100  16.0
396  5.8929 2011-07-30 11.76200  16.0
397  5.8423 2011-08-06 12.56800  16.0
398  5.8163 2011-08-13 16.93200  16.0
399  5.8350 2011-08-20 14.82700  16.0
400  5.8429 2011-08-27 12.14800  16.0
401  5.8970 2011-09-03 17.19700  16.0
402  5.8458 2011-09-10 18.45800  16.0
403  5.9711 2011-09-17 13.07600  16.0
404  5.9550 2011-09-24 13.19300  16.0
405  6.1380 2011-10-01 20.17800  16.0
406  6.0951 2011-10-08 10.92600  16.0
407  5.9874 2011-10-15  9.64180  16.0
408  6.1145 2011-10-22 11.71200  16.0
409  6.1598 2011-10-29  7.32770  16.0
410  6.1276 2011-11-05 11.15600  16.0
411  6.2428 2011-11-12 11.30000  16.0
412  6.2374 2011-11-19 10.77100  16.0
413  7.8198 2011-11-26  6.81090  16.0
414  8.4979 2011-12-03  9.11540  16.0
415  7.0908 2011-12-10  3.52720  16.0
416  6.3069 2011-12-17  0.32245  16.0
417  6.1082 2011-12-24  5.40000  16.0
418  6.1375 2011-12-31 11.93100  16.0
419  3.5696 2011-01-07  0.44508  18.0
420  3.0958 2011-01-14  8.43040  18.0
421  3.8281 2011-01-21 -2.26530  18.0
422  3.7336 2011-01-28  1.33950  18.0
423  3.6466 2011-02-04 10.18700  18.0
424  4.4241 2011-02-11  6.91330  18.0
425  4.7528 2011-02-18  5.30300  18.0
426  4.6170 2011-02-25 10.40700  18.0
427  5.1216 2011-03-04  3.63990  18.0
428  5.3997 2011-03-11  5.49690  18.0
429  5.4961 2011-03-18  2.12060  18.0
430  5.4625 2011-03-25  5.66700  18.0
431  5.4426 2011-04-01 12.13400  18.0
432  5.4857 2011-04-08  6.11440  18.0
433  5.5036 2011-04-15  9.85800  18.0
434  5.5052 2011-04-22 11.39300  18.0
435  5.5101 2011-04-29  9.54510  18.0
436  5.4935 2011-05-06 12.99600  18.0
437  5.5244 2011-05-13  8.83800  18.0
438  5.5015 2011-05-20  8.81900  18.0
439  5.5059 2011-05-27 10.80500  18.0
440  5.5142 2011-06-03 11.08700  18.0
441  5.5896 2011-06-10  7.20970  18.0
442  5.5319 2011-06-17 10.11400  18.0
443  5.5354 2011-06-24 11.29400  18.0
444  5.5589 2011-07-01 11.43900  18.0
445  5.5371 2011-07-08 13.29700  18.0
446  5.5686 2011-07-15 11.61900  18.0
447  5.5670 2011-07-22 12.73800  18.0
448  5.6439 2011-07-29 14.95600  18.0
449  5.6940 2011-08-05 16.48600  18.0
450  5.6338 2011-08-12 16.06000  18.0
451  5.6322 2011-08-19 11.38900  18.0
452  5.6545 2011-08-26 10.69200  18.0
453  5.7226 2011-09-02 14.71100  18.0
454  5.7256 2011-09-09 14.09600  18.0
455  5.7185 2011-09-16 13.03500  18.0
456  5.7454 2011-09-23 11.81800  18.0
457  5.8026 2011-09-30 14.49800  18.0
458  5.8097 2011-10-07 10.76800  18.0
459  5.7557 2011-10-14 13.91800  18.0
460  5.8051 2011-10-21  8.84760  18.0
461  5.8297 2011-10-28  8.37600  18.0
462  5.7679 2011-11-04 12.57400  18.0
463  5.8322 2011-11-11 11.26200  18.0
464  5.8059 2011-11-18 11.11100  18.0
465  5.8491 2011-11-25 12.91700  18.0
466  7.4509 2011-12-02  2.43560  18.0
467  7.1933 2011-12-09  5.87070  18.0
468  6.3294 2011-12-16  2.31850  18.0
469  5.8866 2011-12-23 11.12900  18.0
470  6.1258 2011-12-30  4.78220  18.0
471  3.6297 2011-01-06  2.92880  20.0
472  3.3468 2011-01-13  9.59030  20.0
473  3.8969 2011-01-20 -1.45990  20.0
474  3.8649 2011-01-27  3.49050  20.0
475  3.7189 2011-02-03  6.15450  20.0
476  4.5689 2011-02-10  8.87270  20.0
477  4.8594 2011-02-17  4.61980  20.0
478  4.8374 2011-02-24  7.66730  20.0
479  5.2294 2011-03-03 -0.73976  20.0
480  5.3550 2011-03-10  9.08700  20.0
481  5.5199 2011-03-17  6.57230  20.0
482  5.5185 2011-03-24  4.21580  20.0
483  5.5391 2011-03-31  9.55530  20.0
484  5.5010 2011-04-07 10.24900  20.0
485  5.5107 2011-04-14  8.15550  20.0
486  5.5052 2011-04-21  9.75940  20.0
487  5.5200 2011-04-28  6.43770  20.0
488  5.5096 2011-05-05 10.70900  20.0
489  5.5201 2011-05-12  9.11810  20.0
490  5.5574 2011-05-19  8.26810  20.0
491  5.5375 2011-05-26 11.65000  20.0
492  5.5331 2011-06-02 13.10500  20.0
493  5.5330 2011-06-09 11.09500  20.0
494  5.5517 2011-06-16 12.63000  20.0
495  5.5645 2011-06-23 12.66100  20.0
496  5.5771 2011-06-30  9.57130  20.0
497  5.5920 2011-07-07 13.84300  20.0
498  5.5822 2011-07-14 10.93200  20.0
499  5.6023 2011-07-21 13.44100  20.0
500  5.6408 2011-07-28 13.41700  20.0
501  5.6308 2011-08-04 14.81500  20.0
502  5.6300 2011-08-11 16.33900  20.0
503  5.6460 2011-08-18 10.04300  20.0
504  5.6596 2011-08-25 11.87600  20.0
505  5.6706 2011-09-01 11.71000  20.0
506  5.6735 2011-09-08 11.84600  20.0
507  5.6815 2011-09-15 12.17800  20.0
508  5.6948 2011-09-22 11.90200  20.0
509  5.6955 2011-09-29 17.07200  20.0
510  5.7011 2011-10-06 13.21700  20.0
511  5.7241 2011-10-13 12.01300  20.0
512  5.7566 2011-10-20  3.75340  20.0
513  5.7440 2011-10-27  8.76700  20.0
514  5.7583 2011-11-03 13.96900  20.0
515  5.7599 2011-11-10 13.03200  20.0
516  5.7876 2011-11-17  8.26000  20.0
517  5.7958 2011-11-24 12.02100  20.0
518  5.8690 2011-12-01  9.64050  20.0
519  7.2601 2011-12-08  4.99180  20.0
520  6.4506 2011-12-15  3.73240  20.0
521  5.9143 2011-12-22 12.69300  20.0
522  6.2010 2011-12-29  7.12550  20.0
523  3.7158 2011-01-05  5.66070  22.0
524  3.3825 2011-01-12  4.42930  22.0
525  3.9394 2011-01-19  0.04384  22.0
526  3.8967 2011-01-26  5.33010  22.0
527  3.8006 2011-02-02  4.74030  22.0
528  4.6272 2011-02-09  6.48010  22.0
529  4.8906 2011-02-16  5.92770  22.0
530  4.8437 2011-02-23  7.15490  22.0
531  5.2401 2011-03-02  1.52250  22.0
532  5.2371 2011-03-09  6.46400  22.0
533  5.4725 2011-03-16  6.49500  22.0
534  5.5676 2011-03-23  4.01320  22.0
535  5.5634 2011-03-30 10.23300  22.0
536  5.5158 2011-04-06 12.10300  22.0
537  5.5711 2011-04-13  5.66120  22.0
538  5.5681 2011-04-20  7.94430  22.0
539  5.5749 2011-04-27  9.66080  22.0
540  5.5829 2011-05-04  5.13940  22.0
541  5.5900 2011-05-11 12.00800  22.0
542  5.5885 2011-05-18 13.42900  22.0
543  5.6147 2011-05-25  7.96200  22.0
544  5.6210 2011-06-01  8.94920  22.0
545  5.6379 2011-06-08 10.58700  22.0
546  5.6279 2011-06-15 14.82200  22.0
547  5.6317 2011-06-22 13.28300  22.0
548  5.6570 2011-06-29 10.09900  22.0
549  5.6468 2011-07-06 14.30500  22.0
550  5.6739 2011-07-13 13.44700  22.0
551  5.6677 2011-07-20 13.27900  22.0
552  5.6808 2011-07-27 16.17600  22.0
553  5.6797 2011-08-03 15.40500  22.0
554  5.7022 2011-08-10 11.20800  22.0
555  5.7044 2011-08-17 10.92000  22.0
556  5.7113 2011-08-24 13.57000  22.0
557  5.7130 2011-08-31 12.35700  22.0
558  5.7259 2011-09-07 13.77500  22.0
559  5.7374 2011-09-14 13.05900  22.0
560  5.7435 2011-09-21 11.50800  22.0
561  5.7419 2011-09-28 15.44600  22.0
562  5.7606 2011-10-05 15.41900  22.0
563  5.7488 2011-10-12 12.46600  22.0
564  5.7847 2011-10-19  8.04520  22.0
565  5.7825 2011-10-26  9.85760  22.0
566  5.7979 2011-11-02  9.94480  22.0
567  5.7828 2011-11-09  9.83490  22.0
568  5.8190 2011-11-16  7.18460  22.0
569  5.8245 2011-11-23  3.36370  22.0
570  5.8462 2011-11-30  6.19500  22.0
571  6.8035 2011-12-07  5.57610  22.0
572  6.6398 2011-12-14  5.46150  22.0
573  5.8833 2011-12-21  6.02540  22.0
574  6.0826 2011-12-28  8.87860  22.0
575  3.7401 2011-01-04 -0.87434  24.0
576  3.2508 2011-01-11  7.57030  24.0
577  3.8161 2011-01-18  4.65410  24.0
578  3.8529 2011-01-25  7.24100  24.0
579  3.7638 2011-02-01  2.03130  24.0
580  4.5356 2011-02-08  4.28250  24.0
581  4.6182 2011-02-15  2.09610  24.0
582  4.7871 2011-02-22  4.52120  24.0
583  5.1305 2011-03-01  4.09410  24.0
584  5.2040 2011-03-08 -0.97393  24.0
585  5.4503 2011-03-15  5.77950  24.0
586  5.4623 2011-03-22  4.84410  24.0
587  5.4744 2011-03-29  4.10630  24.0
588  5.4610 2011-04-05 10.80000  24.0
589  5.4958 2011-04-12  7.07030  24.0
590  5.4913 2011-04-19  9.72580  24.0
591  5.4967 2011-04-26 11.22400  24.0
592  5.5384 2011-05-03  7.08340  24.0
593  5.5223 2011-05-10 10.99000  24.0
594  5.5361 2011-05-17 12.02200  24.0
595  5.5549 2011-05-24  9.37830  24.0
596  5.5721 2011-05-31  7.85870  24.0
597  5.5697 2011-06-07  9.30420  24.0
598  5.5709 2011-06-14 10.07000  24.0
599  5.5695 2011-06-21 15.00500  24.0
600  5.5792 2011-06-28 13.74500  24.0
601  5.5783 2011-07-05 17.47700  24.0
602  5.5969 2011-07-12 15.32100  24.0
603  5.6055 2011-07-19 13.27000  24.0
604  5.6135 2011-07-26 11.90700  24.0
605  5.6078 2011-08-02 17.43300  24.0
606  5.6343 2011-08-09 13.26100  24.0
607  5.6386 2011-08-16 13.34400  24.0
608  5.6365 2011-08-23 16.62600  24.0
609  5.6493 2011-08-30 13.30700  24.0
610  5.6607 2011-09-06 14.72500  24.0
611  5.6561 2011-09-13 14.77800  24.0
612  5.6629 2011-09-20 15.70200  24.0
613  5.6921 2011-09-27 11.22700  24.0
614  5.6948 2011-10-04 13.29700  24.0
615  5.6826 2011-10-11 16.68600  24.0
616  5.7206 2011-10-18  7.88400  24.0
617  5.7105 2011-10-25 13.14200  24.0
618  5.6953 2011-11-01 13.24000  24.0
619  5.7325 2011-11-08 10.43400  24.0
620  5.7176 2011-11-15  9.17300  24.0
621  5.7275 2011-11-22  9.06990  24.0
622  5.7293 2011-11-29 12.74600  24.0
623  5.7733 2011-12-06  4.43310  24.0
624  6.7186 2011-12-13  9.05250  24.0
625  5.8346 2011-12-20  7.67070  24.0
626  5.9464 2011-12-27 10.47900  24.0
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5
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What you have is K (5) Groups where you have a dependent (water temp) and an independent series(air temp). This problems is called Pooled Cross-Sectional Time Series Analysis. Construct a separate Transfer Function (ARMAX) model for each of the K groups. Identify a common model (outlier resistant) that would be appropriate. Estimate that model globally using all of the data and then perform an F test to test the hypothesis of a common set of parameters. Upon finding a statistically significant F value examine the coefficients to determine which groups (of the K) that are similar. My current research has been to develop an automatic test for this and we have it operational in a current Beta Version of AUTOBOX (http://www.autobox.com). I would be glad to demonstrate this for you, please post your data. Upon finding out how AUTOBOX conducts this test you might be able to program it yourself or at least have a "destination". Hope this helps.

ADDITIONAL COMMENTS USING KATE'S DATA:

I took the first 4 Groups ( depth of .5,4,6 and 8) and used the first 52 values to construct this example analysis. Following are 4 graphs depicting the Y (water temp) and the X (Air temp) over time for the 4 depths; depth1 ;enter image description here ; depth 2;enter image description here ; depth 3;enter image description here and depth 4;enter image description here . An analysis of the within relationship of Y versus X yielded a typical model of the form enter image description here . I elected to add another AR term to the noise for purposes of a more general expression. All four examples are significantly influenced by anomalies so one might argue/suggest that one should continue with outlier-adjusted series or intervention-scrubbed series i.e. "cleansed series". For presentation purposes here this was not done. We now proceeded to estimate the "typical model" for each of the 4 data sets AND for the composite. This yielded enter image description here. The F test is simply the Chow Test for constant parameters http://en.wikipedia.org/wiki/Chow_test which yielded a significant F. The Chow test simply sums the error sos from each of the 4 cases (each with 52 values) and compares it to the error sos for the composite (208 values). Hope this helps.

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  • $\begingroup$ many thanks for your comments. My data has been added to the question above. $\endgroup$ – KatyB Jul 17 '12 at 16:06
3
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"Similarity" is problem-dependent. From what you write, I have the feeling that estimation of the cross-spectra and coherence might be a good way to proceed, at least if your series are stationary and long enough. Other references that I happen to have handy and may or may not be relevant to your problem are:

Coke, Geoffrey and Tsao, Min (2010). Random effects mixture models for clustering electrical load series. Journal of Time Series Analysis 6: 451-464.

Warren Liao, T (2005). Clustering of time series data - a survey. Pattern Recognition 11: 1857-1874.

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  • 1
    $\begingroup$ Thank you for the editing! I did not know how to format the references. It looks much nicer now. $\endgroup$ – F. Tusell Jul 18 '12 at 17:12
0
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Cross-spectra is just the Fourier transform of the cross-correlation function. The cross-correlation should be high at lag 0 and may drop as the time difference between the series increase. This should be done if both series are stationary. If there are trends or seasonal changes in temperature you could fit time trends or cosine functions to the two series and look for similarities in the model coefficients. For stationary series the same could be done for fitted ARMA models. Do the best fitting series have the same orders p and q? If so how close are the parameter estimates?

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  • $\begingroup$ It is not obvious to me that cross-correlation should be high at lag 0 in this particular case. I would expect temperature in deep water to have very low contemporaneous correlation with air temperature, for instance --even if the weekly observations are averages. It may be the case that correlations at lags different than zero are larger. I would expect air temperature to lead water temperature for relatively shallow depths at the weekly time scale. $\endgroup$ – F. Tusell Jul 18 '12 at 17:06
  • $\begingroup$ The heat content of the well-mixed upper layer of the water should integrate the heat flux from the atmosphere, which includes radiative heat flux from the sun and sensible and latent heat flux with the atmosphere. So the relation between the air and ocean temperature could be quite complex. I recommended looking at the time series carefully to see what your eye picks out before performing any statistical analysis, and also reading up on air-sea heat fluxes and the mixed layer of the ocean. Better yet, talk to someone who works on this. $\endgroup$ – KAE Feb 7 '13 at 18:08

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