1. Underlying single-cycle sinusoidal model
2. Three outliers

and then estimate the constant by the sample mean, then the analysis would suffer because

1. the sample mean would be biased and meaningless;
2. the confidence limits would be meaningless and optimistically small.

(where is the amplitude, is the frequency--between 0 and .5 cycles per observation--, and is the phase) can be fit by standard non-linear least squares, to estimate the coefficients and their uncertainties.

The lag plot is also of value in outlier-detection. Note in the above plot, that there appear to be 4 points lying off the ellipse. However, in a lag plot, each point in the original data set Y shows up twice in the lag plot--once as Y(i) and once as Y(i-1). Hence the outlier in the upper left at Y(i) = 300 is the same raw data value that appears on the far right at Y(i-1) = 300. Thus (-500,300) and (300,200) are due to the same outlier, namely the 158th data point: 300. The correct value for this 158th point should be approximately -300 and so it appears that a sign got dropped in the data collection. The other 2 points lying off the ellipse: at roughly (100,100) and at (0,-50) are caused by 2 faulty data values: the third data point of -15 should be about +125 and the fourth data point of +141 should be about -50, respectively. Hence the 4 apparent lag plot outliers are traceable to 3 actual outliers in the original run sequence: at points 4 (-15), 5 (141) and 158 (300). In retrospect, only one of these (point 158 (= 300)) is an obvious outlier in the run sequence plot.

1. Do a spectral plot to obtain an initial estimate of the frequency of the underlying cycle. This will be helpful as a starting value for the subsequent non-linear fitting.
2. Omit the 3 outliers.
3. Carry out a non-linear fit of the model to the 197 points.