title: How to fit a function that is an exponential function times a sum of two sinusoidal functions
body: <p>I am having difficulty getting <code>nls</code> or <code>nlsLM</code> to fit a function to the following time series in R.</p>
  
  <pre><code>  [1] 143.7083 143.7083 141.5833 139.8750 138.8333 137.9167 136.4583 134.8750 134.3750
   [10] 134.1250 133.7917 132.5000 129.2083 127.2917 127.9583 129.1667 129.8333 130.0000
   [19] 130.5000 131.1667 131.8750 132.2500 131.5417 132.1667 133.5417 133.7500 133.7500
   [28] 134.2083 134.7083 135.2500 136.2083 136.7083 136.7500 135.2083 134.4167 134.6667
   [37] 134.6667 135.4583 136.3750 136.5000 137.3750 138.2083 137.4167 137.0833 137.0417
   [46] 138.2500 139.5000 140.0417 139.1667 138.0833 137.8750 138.1250 137.5000 136.0417
   [55] 135.7917 135.2083 135.4583 136.5000 136.5417 135.2083 133.7083 132.0000 131.4583
   [64] 131.9167 132.4167 132.2500 132.2500 134.1250 134.7083 133.7083 133.0833 131.9167
   [73] 130.8750 131.7500 131.5833 129.4167 127.4167 127.4583 127.3750 125.7500 124.7500
   [82] 124.6667 125.5833 129.4583 132.2917 131.8333 132.2083 134.2917 136.7500 138.2917
   [91] 139.0000 138.5417 138.9167 138.5000 137.3333 136.3750 135.4167 133.4167 130.5000
  [100] 128.8333 127.6667 125.8750 125.0833 126.1667 125.8750 126.0000 127.1667 124.6250
  [109] 122.2917 124.4167 126.9583 128.2083 129.4167 131.5000 132.2500 132.6250 134.1250
  [118] 135.5000 135.4583 137.5833 141.1667 142.0000 139.6667 138.0417 137.7917 136.2917
  [127] 135.2500 134.4583 133.8750 132.8333 132.7083 131.1250 127.3750 126.1250 129.5417
  [136] 130.6667 128.4167 127.9167 128.6250 129.1250 129.8750 131.0833 131.2083 132.3333
  [145] 134.5000 134.5833 132.5000 132.2083 134.0833 135.2500 134.1667 132.7083 130.5417
  [154] 128.5833 127.4167 125.5000 124.0000 122.2500 122.5833 124.0000 123.1250 122.2500
  [163] 122.8333 123.1667 122.6667 122.6250 123.9167 124.6250 125.6667 128.1250 128.5000
  [172] 127.2500 127.3750 128.1250 128.3333 128.3750 129.0417 128.7500 127.7500 130.1667
  [181] 131.2500 130.3333 129.4583 127.8333 127.6250 127.0417 125.8333 125.8750 126.8750
  [190] 127.7917 127.7083 126.7917 126.0833 125.4583 126.5000 128.0000 127.0833 126.5000
  [199] 127.4167 128.3750 128.2917 127.8750 128.4583 128.5833 130.3750 132.2500 131.0833
  [208] 130.9167 130.7917 130.1250 130.3333 130.0833 129.0833 128.1667 126.2083 124.5417
  [217] 122.4167 119.4583 119.9583 121.3333 122.9583 125.3333 126.7500 127.6667 128.5417
  [226] 129.7500 132.2083 133.1667 132.7083 133.6250 132.2917 130.4583 130.3750 129.9583
  [235] 129.2917 128.2917 128.0833 127.8750 126.2500 125.3750 124.5417 123.7917 124.2917
  [244] 124.0417 123.9583 123.7500 122.9167 122.7083 123.4583 125.1250 127.0417 128.1250
  [253] 128.8333 130.0417 131.0833 131.0417 129.7917 129.4167 129.8750 129.7917 129.7500
  [262] 130.9167 131.8750 133.7500 137.7083 139.3333 139.5000 140.3750 141.2917 140.9167
  [271] 140.2500 140.5000 140.1667 137.5000 135.1250 135.1667 134.7917 134.3750 136.0000
  [280] 138.3333 138.5000 138.9167 139.4583 139.2917 139.5833 140.2917 140.9583 139.5833
  [289] 139.1250 139.5833 137.9583 136.6667 136.7500 135.7083 134.5833 134.4583 133.5833
  [298] 132.0000 130.9167 130.7500 130.2917 130.8750 131.2917 129.1250 128.4167 130.1667
  [307] 131.1250 131.6667 132.4583 134.3333 137.2500 138.5000 137.1250 137.2917 140.1667
  [316] 143.3750 144.4167 144.1250 144.8333 144.5833 143.8750 142.0417 137.7083 133.7083
  [325] 130.1250 125.3750 122.7917 122.5417 123.0000 122.7917 122.7083 124.5000 126.9583
  [334] 129.1250 132.5833 136.3750 139.8333 140.2917 137.0417 135.3750 134.9583 135.6250
  [343] 135.7917 135.3750 134.4167 132.8750 131.7917 131.0833 131.2917 131.2917 132.0417
  [352] 132.6250 132.4583 131.2917 129.4583 127.8333 127.0000 127.4583 128.1667 129.4167
  [361] 130.5417 132.4583 133.0000 132.8750 133.8750 134.5833 135.9167 137.4167 138.5833
  [370] 139.0833 137.5000 134.5833 131.3333 131.0833 132.4167 132.1250 131.2083 131.0833
  [379] 131.2917 129.1250 126.5833 125.3750 125.7500 126.5833 127.1250 126.9583 127.2083
  [388] 126.7500 126.7917 127.2917 127.0000 127.8333 128.4167 128.9167 129.8333 131.9583
  [397] 135.0833 136.5417 137.0000 137.7500 136.7083 135.3750 135.2083 135.9167 137.0417
  [406] 138.4167 139.7500 139.0833 137.5417 138.9583 139.8333 139.0000 139.5833 140.7500
  [415] 141.2083 140.7917 139.9167 139.7500 139.1250 139.2500 139.5000 136.3333 135.5000
  [424] 137.7917 139.5000 139.1250 138.4167 138.3333 138.6667 138.6667 140.2500 140.2083
  [433] 139.0417 139.9583 139.5000 138.1667 137.0833 137.7083 138.9583 139.1667 138.6250
  [442] 138.1667 136.0417 135.1250 135.9167 135.2500 134.2917 133.5000 132.8750 132.9167
  [451] 133.0000 133.4167 134.4167 133.2083 131.9583 132.5833 132.2500 132.5833 133.6667
  [460] 133.9583 133.7917 133.1250 132.7083 133.2500 132.9583 132.0417 131.9167 131.0000
  [469] 132.2083 134.1667 134.1667 134.3750 134.3333 134.9583 135.1250 133.7917 133.9167
  [478] 136.4583 137.5833 134.7083 133.1667 135.5000 137.6250 138.4583 138.5833 137.7917
  [487] 137.7083 138.2917 137.4167 136.7917 137.7500 142.2500 144.8750 142.8750 141.8750
  [496] 142.1250 144.0833 145.5000 145.2917 145.4167 145.9583 145.5000 144.4167 143.4167
  [505] 142.5833 142.0833 140.7917 139.1667 136.5000 133.6250 132.5417 131.4583 131.0000
  [514] 130.5833 129.9167 129.8333 130.0833 129.1667 127.4167 127.0833 128.1667 130.1250
  [523] 131.8333 133.5833 134.5417 134.9583 135.1250 133.9583 131.0833 129.9167 131.6250
  [532] 134.0000 134.4167 134.1667 134.9583 134.2500 133.0417 132.5000 133.2917 135.9583
  [541] 139.8333 141.8333 142.2083 141.7500 141.8333 142.2917 141.7500 142.7917 144.4167
  [550] 146.0417 145.0417 140.7083 137.0833 135.7917 134.0833 131.9583 130.9167 130.1250
  [559] 128.6667 126.3333 125.0417 124.2083 123.4167 125.0833 127.3333 129.1250 132.0000
  [568] 132.9167 132.1250 131.1250 131.0000 131.7083 131.8750 132.1250 133.7500 134.7917
  [577] 136.1250 138.0000 138.3333 138.9583 139.8750 140.5833 140.6250 140.9583 142.3750
  [586] 143.2500 143.1250 143.3750 143.1667 141.8333 140.2083 139.4583 139.9583 140.4583
  [595] 141.4583 142.3333 141.6250 140.4583 140.9583 141.7917 140.2917 139.0833 140.7083
  [604] 142.0417 142.2917 143.2083 142.1667 140.1250 139.1667 139.0833 137.8750 134.3750
  [613] 132.9167 133.2083 131.4167 129.9583 129.3750 128.1667 127.1667 126.7500 126.0833
  [622] 124.4167 124.6250 127.7500 130.3333 130.5833 131.1667 132.3333 132.4167 131.9583
  [631] 132.9167 134.2917 135.6250 137.4583 137.2917 134.9583 131.2500 127.6667 126.2500
  [640] 125.1667 123.4583 122.4167 121.9583 121.7083 121.6250 120.8750 121.2917 122.7917
  [649] 123.1667 123.6667 124.0000 124.4167 127.4167 129.8750 129.8333 129.2500 128.4583
  [658] 128.4167 127.7500 127.0833 128.5417 130.1667 129.1250 127.4167 124.2083 121.6250
  [667] 122.2500 123.2917 124.2500 125.1667 126.2083 125.8333 126.7500 128.2500 128.9167
  [676] 130.0833 131.4167 132.2500 132.0833 131.8750 130.6667 129.0000 128.7083 130.7500
  [685] 131.8333 129.8750 128.8750 128.5833 128.0417 128.2083 127.7917 127.3333 127.9167
  [694] 130.3750 130.9167 128.8750 127.1250 128.2500 130.7500 131.9583 132.5417 133.0417
  [703] 134.0000 133.2500 132.6250 130.8333 128.7917 129.5417 129.2917 128.0000 126.9583
  [712] 126.1667 127.0417 127.4167 126.6250 127.8750 129.0000 129.6250 132.1667 132.8750
  [721] 132.5417 131.6667 131.4167 133.2917 134.5000 136.4583 139.1250 140.2500 141.0417
  [730] 142.0417 141.2917 138.9167 137.0000 138.5833 139.8333 139.4583 140.0833 140.0000
  [739] 140.0417 140.3750 140.7083 141.0833 142.2500 146.5417 151.9583 154.7500 157.3750
  [748] 158.8333 158.5833 158.5000 158.2917 159.0417 158.9583 159.0000 158.0833 156.9167
  [757] 156.9167 155.7917 152.2917 149.7917 149.2500 149.0000
  </code></pre>
  
  <p>I have an equation that looks like fits it pretty well when plotted against the time series.</p>
  
  <pre><code>(-6.1253+0.0859744+2.14215 + -6.1253*0.0859744*-7.19365e-10*1.00001^(10600.6546*
  ((1:762) + -510.4607))) * (sin((((1:762) + 4624.28)/3699.41)^7.11555 + -5.20009) + 
  sin((((1:762) + -0.873172)/48.5129)^1.06537 + -9.29727))
  </code></pre>
  
  <p>When I use <code>nls</code> or <code>nlslm</code> and use the parameters it gives to plot the equation against the data, the equation strays from the data near the end where the exponential behavior becomes significant, even when I only leave one of the numbers as a parameter.</p>
  
  <pre><code>mntm_trend_fit_sin_exp=nlsLM(mntm_trend_num ~ 133.495+0.909436 + 
  (-6.1253+0.0859744+2.14215 + -6.1253*0.0859744*-7.19365e-10*1.00001^(inexpco*((1:762) + 
  -510.4607))) * (sin((((1:762) + 4624.28)/3699.41)^7.11555 + -5.20009) + sin((((1:762) + 
  -0.873172)*48.5129)^1.06537 + -9.29727)), start=list(inexpco=10600.6546), trace=TRUE)
  </code></pre>
  
  <p><code>mntm_trend_num</code> is the posted time series.</p>
  
  <p>The aforementioned function has a significantly better RMSE and MAE, yet 'nls' and 'nlslm' strays away from my starting point of <code>inexpco=10600</code>.</p>
  
  <p>Any and all suggestions are invited.</p>
  
  <p>Thank you in advance.</p>