trend2d

trend2d(cmd0::String="", arg1=nothing, kwargs...)

Fit [weighted] [robust] polynomial for z = f(x,y) to xyz[w] data.

Description

trend2d reads x,y,z [and w] values from the first three [four] columns [or xyz[w]file] and fits a regression model z = f(x,y) + e by [weighted] least squares. The fit may be made robust by iterative reweighting of the data. The user may also search for the number of terms in f(x,y) which significantly reduce the variance in z. n_model may be in [1,10] to fit a model of the following form (similar to grdtrend):

\[z(x,y) = m_1 + m_{2}x + m_{3}y + m_{4}xy + m_{5}x^2 + m_{6}y^2 + m_{7}x^3 + m_{8}x^{2}y + m_{9}xy^2 + m_{10}y^3\]

The user must specify model=n, the number of model parameters to use; thus, fits a bilinear trend, model=6 a quadratic surface, and so on. Optionally, append +r to perform a robust fit. In this case, the program will iteratively reweight the data based on a robust scale estimate, in order to converge to a solution insensitive to outliers. This may be handy when separating a "regional" field from a "residual" which should have non-zero mean, such as a local mountain on a regional surface.

Required Arguments

• table
  Data table(s) containing x,y,z [w] values in the first 3 [4] columns.

• F or out or output : – out=:xyzmrw | out=:p
  Specify up to six letters from the set {x y z m r w} in any order to create columns of output. x = x, y = y, z = z, m = model f(x,y), r = residual z - m, w = weight used in fitting. Alternatively, to just report the model parameters, use out=:p.

• N or model : – model=n | model="n+r"
  Specify the number of terms in the model, n_model, and use model="n_model+r" to do a robust fit. E.g., a robust bilinear model is model="4+r".

Optional Arguments

• C or condition_number : – condition_number=number
  Set the maximum allowed condition number for the matrix solution. trend2d fits a damped least squares model, retaining only that part of the eigenvalue spectrum such that the ratio of the largest eigenvalue to the smallest eigenvalue is condition_#. [Default: condition_# = 1.0e06].

• I or conf_level : – conf_level=true | conf_level=level
  Iteratively increase the number of model parameters, starting at one, until n_model is reached or the reduction in variance of the model is not significant at the confidence_level level. You may set conf_level=true only, without an attached number; in this case the fit will be iterative with a default confidence level of 0.51. Or choose your own level between 0 and 1. See remarks section.

• V or verbose : – verbose=true | verbose=level
  Select verbosity level. More at verbose

• W or weights : – weights=true | weights="+s|+w"
  Weights are supplied in input column 4. Do a weighted least squares fit [or start with these weights when doing the iterative robust fit]. Append +s to instead read data uncertainties (one sigma) and create weights as 1/sigma^2, or use the weights as read (+w) [Default reads only the first 2 columns].

• bi or binary_in : – binary_in=??
  Select native binary format for primary table input. More at

• bo or binary_out : – binary_out=??
  Select native binary format for table output. More at

• di or nodata_in : – nodata_in=??
  Substitute specific values with NaN. More at

• e or pattern : – pattern=??
  Only accept ASCII data records that contain the specified pattern. More at

• f or colinfo : – colinfo=??
  Specify the data types of input and/or output columns (time or geographical data). More at

• g or gap : – gap=??
  Examine the spacing between consecutive data points in order to impose breaks in the line. More at

• h or header : – header=??
  Specify that input and/or output file(s) have n header records. More at

• i or incol or incols : – incol=col_num | incol="opts"
  Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only). More at incol

• q or inrows : – inrows=??
  Select specific data rows to be read and/or written. More at

• s or skiprows or skip_NaN : – skip_NaN=true | skip_NaN="<cols[+a][+r]>"
  Suppress output of data records whose z-value(s) equal NaN. More at

• w or wrap or cyclic : – wrap=??
  Convert input records to a cyclical coordinate. More at

• yx : – yx=true
  Swap 1st and 2nd column on input and/or output. More at

Remarks

The domain of x and y will be shifted and scaled to [-1, 1] and the basis functions are built from Chebyshev polynomials. These have a numerical advantage in the form of the matrix which must be inverted and allow more accurate solutions. In many applications of trend2d the user has data located approximately along a line in the x,y plane which makes an angle with the x axis (such as data collected along a road or ship track). In this case the accuracy could be improved by a rotation of the x,y axes. trend2d does not search for such a rotation; instead, it may find that the matrix problem has deficient rank. However, the solution is computed using the generalized inverse and should still work out OK. The user should check the results graphically if trend2d shows deficient rank. NOTE: The model parameters listed with verbose are Chebyshev coefficients; they are not numerically equivalent to the m#s in the equation described above. The description above is to allow the user to match model with the order of the polynomial surface. For evaluating Chebyshev polynomials, see grdmath.

The model="n_model+r" (robust) and conf_level (iterative) options evaluate the significance of the improvement in model misfit Chi-Squared by an F test. The default confidence limit is set at 0.51; it can be changed with the conf_level option. The user may be surprised to find that in most cases the reduction in variance achieved by increasing the number of terms in a model is not significant at a very high degree of confidence. For example, with 120 degrees of freedom, Chi-Squared must decrease by 26% or more to be significant at the 95% confidence level. If you want to keep iterating as long as Chi-Squared is decreasing, set conf_level to zero.

A low confidence limit (such as the default value of 0.51) is needed to make the robust method work. This method iteratively reweights the data to reduce the influence of outliers. The weight is based on the Median Absolute Deviation and a formula from Huber [1964], and is 95% efficient when the model residuals have an outlier-free normal distribution. This means that the influence of outliers is reduced only slightly at each iteration; consequently the reduction in Chi-Squared is not very significant. If the procedure needs a few iterations to successfully attenuate their effect, the significance level of the F test must be kept low.

Examples

To remove a planar trend from data.xyz by ordinary least squares, use

D = trend2d("data.xyz", out=:xyr, model=3)

To simply report the three coefficients, use

trend2d("data.xyz", out=:p, model=3)

To make the above planar trend robust with respect to outliers, use

D = trend2d("data.xyz", out=:xyr, model="3+r")

To find out how many terms (up to 10 in a robust interpolant are significant in fitting data.xyz, use

trend2d("data.xyz", model="10+r", conf_level=true, verbose=true)

See Also

grdtrend, trend1d

References

Huber, P. J., 1964, Robust estimation of a location parameter, Ann. Math. Stat., 35, 73-101.

Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory, Revised Edition, Academic Press, San Diego.