Upload
muhammad-taufiq
View
221
Download
1
Tags:
Embed Size (px)
DESCRIPTION
fdgf
Citation preview
GEOKOMPUTASIPertemuan ke-6
GEOKOMPUTASI
Curve Fitting and Optimization
Introduction
Least Square Regression
Linear
Non-Linear
Curve Fitting and Optimization
Introduction
Data is often given for discrete values along a continuum. However, you may require estimates at points between the discrete values
Describes techniques to fit curves to such data to obtain intermediate estimatesestimates
General approaches for curve fitting : where the data exhibits a significant degree of error or noise, the strategy is to
derive a single curve that represents the general trend of the datathis nature is called least-squares regression
where the data is known to be very precise, the basic approach is to fit a curveseries of curves that pass directly through each of the pointsvalues between well-known discrete points is called
Data is often given for discrete values along a continuum. However, estimates at points between the discrete values
techniques to fit curves to such data to obtain intermediate
eneral approaches for curve fitting :where the data exhibits a significant degree of error or noise, the strategy is to derive a single curve that represents the general trend of the data. One approach of
squares regressionwhere the data is known to be very precise, the basic approach is to fit a curve or a series of curves that pass directly through each of the points. The estimation of
points is called interpolation
Three attempts to fit a best curve through five data points.
(a) Least-squares regression, (b) linear interpolation,
Three attempts to fit a best curve through five data
b) linear interpolation, (c) curvilinear interpolation
Two types of applications are generally encountered when fittingexperimental data: trend analysis and hypothesis testing.
Trend Analysis :represents the process of using the patternUsed to predict or forecast values of the dependent variableUsed to predict or forecast values of the dependent variable
Hypothesis Testing :an existing mathematical model is compared with measured datamodel coefficients are unknown, it may be necessary to determine values that best fit the observed data. On the other hand, if estimates of the model coefficients are already available, it mayvalues of the model with observed values to test theOften, alternative models are compared and the best one is selectedbasis of empirical observations.
Two types of applications are generally encountered when fittingexperimental data: trend analysis and hypothesis testing.
pattern of the data to make predictions. to predict or forecast values of the dependent variableto predict or forecast values of the dependent variable
is compared with measured data. If the are unknown, it may be necessary to determine values that
data. On the other hand, if estimates of the model coefficients are already available, it may be appropriate to compare predicted values of the model with observed values to test the adequacy of the model. Often, alternative models are compared and the best one is selected on the
Least Square Regression
(a) Data exhibiting significant
error,
(b) Polynomial fit
oscillating beyond the range of
the data
(b) Polynomial fit
oscillating beyond the range of(c) More satisfactory
result using the least-squares fit.
Linear Regression
The simplest example of a least-square approximation is fitting a straight line paired observation : (x1, y1), (x2, y2), . . . , (
y = a0 + a1x + e a0 = coefficients representing the intercept
a coefficients representing the slope a1 = coefficients representing the slope
e = error, or residual between the model and the observations
e = y - a0 - a1x
The error, or residual, is the discrepancy between the true value of y and the approximate value, a0 + a1x , predicted by the linear equation
square approximation is fitting a straight line paired ), . . . , (xn, yn).
coefficients representing the intercept
coefficients representing the slopecoefficients representing the slope
error, or residual between the model and the observations
he error, or residual, is the discrepancy between the true value of y and the predicted by the linear equation
Linear Regression
Criteria for a "Best" Fit
minimize the sum of the residual errors for all the available data
(a) minimizes the sum of the residuals,
minimize the sum of the residual errors for all the available data
of the residuals,
Linear Regression
Criteria for a "Best" Fit
(b) minimizes the sum of the absolute
values of the residuals,
b) minimizes the sum of the absolute
values of the residuals,
Linear Regression
Criteria for a "Best" Fit A third strategy for fitting a best line is the
technique, the line is chosen that minimizes the maximum distance that an individual point falls from the line
It should be noted that the minimax principle is sometimes wellfitting a simple function to a complicatedWilkes, 1969).
(c) Minimizes the maximum error of any individual point
third strategy for fitting a best line is the minimax criterion. In this is chosen that minimizes the maximum distance that an
principle is sometimes well-suited for fitting a simple function to a complicated function (Carnahan, Luther, and
Linear Regression
Minimize the sum of the squares of the residuals between the measured y and the y calculated with the linear model
the sum of the squares of the residuals between the with the linear model
Linear Regression
Least-Squares Fit of a Straight Line
To determine values for a0 + a1
Setting these derivatives equal to zero will result
in a minimum Sr
Squares Fit of a Straight Line
Setting these derivatives equal to zero will result
realizing that a0 = na0
where y and x are the means of y and x, respectively.y and x are the means of y and x, respectively.
Example
Fit a straight line to the x and y values in the first two columns
SOLUTION
Xi Yi Xi^2
1 0.5 1
2 2.5 4
3 2 9
4 4 164 4 16
5 3.5 25
6 6 36
7 5.5 49
jml X 28 jml Y 24 jml Xi^2 140
rata X 4 rata Y 3.428571
Quantification of Error of Linear Regression
the square of the residual represented the square of the discrepancy between the data and a single estimatebetween the data and a single estimatetendencythe mean
the square of the residual represents the square of the vertical distance between the data and another measure ofstraight line
Quantification of Error of Linear Regression
of the residual represented the square of the discrepancy between the data and a single estimate of the measure of central between the data and a single estimate of the measure of central
represents the square of the vertical distance between the data and another measure of central tendencythe
Regression data showing (a) the spread of the data around the mean of the dependent variable
data around the best-fit line. The reduction in the spread in going from
the right, represents the improvement due to linear regression
a) the spread of the data around the mean of the dependent variable and (b) the spread of the
fit line. The reduction in the spread in going from (a) to (b), as indicated by the bell-shaped curves at
linear regression
Polynomial Regression
For example, suppose that we fit a secondquadraticFor example, suppose that we fit a second-order polynomial or
where all summations are from i = 1 through n. Note that the above three equations are linear and have three unknowns: The coefficients of the unknowns can be calculatedobserved data.
The two-dimensional case can be easily extended to an polynomial as
the standard error is formulated asmth
= 1 through n. Note that the above and have three unknowns: a0, a1, and a2.
The coefficients of the unknowns can be calculated directly from the
dimensional case can be easily extended to an mth-order
mth-order polynomial
EXAMPLE
Fit a second-order polynomial to the data in the first
two columns
order polynomial to the data in the first
SOLUTION
Solving these equations through a technique such as Gauss
elimination gives
a1 = 2.35929, and a2 = 1.86071. Therefore, the least
quadratic equation for this
case is y = 2.47857 + 2.35929x + 1.86071x2
The standard error of the estimate based on the regression
polynomial is
Solving these equations through a technique such as Gauss
elimination gives a0 = 2.47857,
a1 = 2.35929, and a2 = 1.86071. Therefore, the least-squares
quadratic equation for this
y = 2.47857 + 2.35929x + 1.86071x2
The standard error of the estimate based on the regression