I have water temperature data consisting of monthly means for 20 years. As one would expect there is a definite seasonal/cyclical pattern. I wish to model the time series data by fitting an ARIMA model.
Two questions:
Would this be an appropriate analysis for temperature data?
How do I test if there is a significant trend (a p-value would be nice!) of increasing water temperature in this data which is clearly nonlinear?
Year Month Temperature 1953 March 21.88302419 1953 April 21.22354167 1953 May 19.98760753 1953 June 18.39943056 1953 July 16.94803763 1953 August 16.71372312 1953 September 17.46852778 1953 October 18.12665323 1953 November 19.29626389 1953 December 19.9861828 1954 January 20.86797043 1954 February 21.91311012 1954 March 22.3521371 1954 April 20.94972222 1954 May 20.34298387 1954 June 18.39666667 1954 July 17.18486559 1954 August 17.13916667 1954 September 17.35176389 1954 October 18.30989247 1954 November 19.14590278 1954 December 20.25358871 1955 January 21.8128629 1955 February 22.73234195 1955 March 21.60393817 1955 April 20.92319444 1955 May 20.09857527 1955 June 18.16069444 1955 July 17.24068641 1955 August 17.14547043 1955 September 16.676875 1955 October 17.31141129 1955 November 19.70297222 1955 December 20.0072043 1956 January 21.77598118 1956 February 21.62849702 1956 March 21.94706989 1956 April 20.6965 1956 May 19.05202957 1956 June 17.81277778 1956 July 17.3078629 1956 August 17.35629032 1956 September 17.84531944 1956 October 18.0919086 1956 November 19.68886111 1956 December 20.42611559 1957 January 21.82801075 1957 February 22.76324405 1957 March 22.8733109 1957 April 21.89454167 1957 May 20.41923387 1957 June 18.85397222 1957 July 18.36353495 1957 August 17.7866218 1957 September 18.141875 1957 October 18.99646505 1957 November 19.38388889 1957 December 20.64235215 1958 January 21.35995968 1958 February 21.56925595 1958 March 23.04430108 1958 April 21.3755 1958 May 20.06209677 1958 June 18.8416968 1958 July 17.77298387 1958 August 17.71418011 1958 September 17.39784722 1958 October 17.88056452 1958 November 19.88647222 1958 December 21.30474462 1959 January 22.51991925 1959 February 23.35799107 1959 March 22.5344086 1959 April 22.08238889 1959 May 20.04969086 1959 June 19.00790278 1959 July 18.27353495 1959 August 17.55283984 1959 September 17.22833333 1959 October 18.01680108 1959 November 18.6775 1959 December 19.8469086 1960 January 20.68198925 1960 February 21.83154762 1960 March 21.50378073 1960 April 20.34733796 1960 May 19.58559588 1960 June 18.45406456 1960 July 17.53125 1960 August 16.96117944 1960 September 16.98905903 1960 October 17.73582997 1960 November 18.65800694 1960 December 19.32586694 1961 January 19.71767473 1961 February 21.04120651 1961 March 21.35288306 1961 April 20.30002431 1961 May 18.6331754 1961 June 18.62307639 1961 July 18.2453629 1961 August 16.99882056 1961 September 17.33630194 1961 October 16.92414798 1961 November 18.00218056 1961 December 18.73533154 1962 January 19.35255376 1962 February 20.21777565 1962 March 21.21165323 1962 April 20.21122222 1962 May 19.51232527 1962 June 18.25903472 1962 July 17.53709677 1962 August 16.88536713 1962 September 16.02052778 1962 October 17.28112903 1962 November 18.13883333 1962 December 19.95607527 1963 January 21.23651882 1963 February 19.9141523 1963 March 21.89069892 1963 April 21.20695833 1963 May 19.74677419 1963 June 18.01259722 1963 July 17.27166667 1963 August 16.63431452 1963 September 16.78509722 1963 October 18.14424731 1963 November 19.29784722 1963 December 18.66950269 1964 January 22.3822043 1964 February 21.98738095 1964 March 22.06516129 1964 April 20.77536364 1964 May 19.60976277 1964 June 17.69825347 1964 July 17.47606586 1964 August 16.81932997 1964 September 17.18971597 1964 October 17.78017742 1964 November 19.022725 1964 December 19.98706116 1965 January 20.62890995 1965 February 21.86117039 1965 March 21.76144624 1965 April 21.41317694 1965 May 19.89478226 1965 June 18.14042639 1965 July 16.53863575 1965 August 17.4485 1965 September 17.38953973 1965 October 17.79755645 1965 November 19.54316042 1965 December 20.5874375 1966 January 21.33427016 1966 February 22.30857515 1966 March 21.52931855 1966 April 21.37102778 1966 May 20.02285954 1966 June 18.68837361 1966 July 18.03406653 1966 August 17.54710551 1966 September 18.25538665 1966 October 18.1833172 1966 November 19.20074583 1966 December 20.33642742 1967 January 21.07148185 1967 February 22.28252799 1967 March 21.35484005 1967 April 20.18232083 1967 May 18.87129435 1967 June 18.12438472 1967 July 17.34492215 1967 August 17.27377688 1967 September 16.92026389 1967 October 17.54733871 1967 November 18.37069444 1967 December 19.83104839 1968 January 21.33081989 1968 February 21.68643155 1968 March 21.64960081 1968 April 20.69278889 1968 May 18.98352957 1968 June 18.3307 1968 July 17.23593011 1968 August 16.94599866 1968 September 16.48676389 1968 October 16.9316371 1968 November 17.66956806 1968 December 19.41569489 1969 January 21.37015591 1969 February 20.70526488 1969 March 21.64435081 1969 April 20.97368472 1969 May 20.4197836 1969 June 18.31870833 1969 July 17.4229328 1969 August 17.10003226 1969 September 17.36448056 1969 October 18.07489785 1969 November 18.97591528 1969 December 20.51862231 1970 January 22.03919489 1970 February 21.64364487 1970 March 21.24479032 1970 April 20.28298611 1970 May 19.29274059 1970 June 17.73382639 1970 July 16.55055511 1970 August 16.43872446 1970 September 17.48788472 1970 October 17.9552638 1970 November 19.17982222 1970 December 19.68649597 1971 January 21.57199866 1971 February 21.76181178 1971 March 21.11755511 1971 April 20.01336667 1971 May 19.05388978 1971 June 18.52273333 1971 July 17.39878226 1971 August 16.3497836 1971 September 17.0700125 1971 October 18.32019758 1971 November 18.96098333 1971 December 19.1836586
was developed BUT a statistically significant change point was detected using the CHOW Test 
. The most recent 79 values were then examined to identify a suitable model.
withe the following Actual/Fit and Forecast graph. 
. The residuals from this model suggest randomness suggesting a sufficient model. 
. The forecast plot/table is presented here 
. Finally the Actual/Cleansed graph showing the two anomalous points is interesting and informative 
. The statistical summary is here 
and here 
There is no suggestion in the data that would support a "trend" conclusion. Since there is no evidence of a transient error variance or a level shift in the model's errors , any number of simple/uncomplicated approaches should work with data if you ignore the change in parameters over time .
and the periods 97 - 120 
and finally the last set (175-200) 
