MATLAB MATRIX LABORATORY

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????? ??????

• ???? ?????? ???????
• ????? ?????? BASIC FITTING
• 2 ????? ?????? Curve Fitting Toolbox (CF TBX)
• 3 ????? ?????? ?? ??? DISTRIBUTION FITTING TOOL

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????? ????? ?????? (Fitting)
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• Function approximation
• -- method of least squares
• -- minimize the residuals
• -- given data of points have noises
  (uncertainty)
• the purpose is to find the trend
  represented by data.
• Function interpolation
• -- approximating function match the given data
  exactly
• -- given data of points are precise
• the purpose is to find data between these
  points

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????? ????? ??????(???? 2)
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• Minimizing Residuals (Errors)
• minimum average error (cancellation)
• minimum absolute error
• minimal error (minimizing the maximum error)
• least squares (linear, quadratic, .)

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????? ????? ?????? (???? 4)
?????? ???????? Fitting a straight line to
observations Equation for straight line
Difference between observation and line ei is
the residual or error
Small residual errors Large residual errors
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• Minimize total square-error function sum of the
  squares of the residuals - Sr(a0 ,a1)
• Solve for (a0
  ,a1)

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• Minimize

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• Positive differences do not cancel negative
  differences
• Differentiation is straightforward
• Weighted differences
• Small differences become smaller and large
  differences are magnified

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??????? ??????? - ?????

• Polynomial fit for
• error function - erf(x)
• x (0 0.1 2.5)'
• y erf(x)
• p polyfit(x,y,6)
• f polyval(p,x)
• table x y f y-fy-f absolute error
• x (0 0.1 5)'
• y erf(x)
• f polyval(p,x)
• plot(x,y,'o',x,f,'-')
• axis(0 5 0 2)

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• Use sum( ) in MATLAB

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• Sum of squares of the residuals with respect to
  the mean
• Sum of squares of the residuals with respect to
  the regression line
• Coefficient of determination
• Correlation coefficient

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• Alternative formulation of correlation
  coefficient
• More convenient for computer implementation

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????? ?????? BASIC FITTING

• ????? ?????? ???? ??
• ???? ????? ???? ???? Basic Fitting
• ???????? ???????
• ???? ????? ??????

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• Fit data using a spline interpolant, a
  shape-preserving interpolant, or a polynomial up
  to degree 10.
• Plot multiple fits simultaneously for a given
  data set.
• Plot the fit residuals.
• Examine the numerical results of a fit.
• Evaluate (interpolate or extrapolate) a fit.
• Annotate the plot with the numerical fit results
  and the norm of residuals.
• Save the fit and evaluated results to the MATLAB
  workspace.

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• 1.Plot the data US Population (in millions of
  people)
• 1790 3.9
• 1800 5.3
• 1810 7.2
• 1820 9.6
• 1830 12.9
• 1840 17.1
• 1850 23.1
• 1860 31.4
• 1870 38.6
• 1880 50.2
• 1890 62.9
• 1900 76
• 1910 92
• 1920 105.7
• 1930 122.8
• 1940 131.7
• 1950 150.7
• 1960 179

2.Select Basic Fitting from the Tools menu of the
created plot.
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3.Press here and then press here
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4. Sign the needed places, for instance 4th
degree polynomial ,Show equation , Plot residuals
Possible message
Sign Center and scale X data
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6. You can specify a vector of x-values at which
to evaluate the new value of population. Enter
needed value (f.e. 2005) and then click Evaluate
and Plot evaluated results
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7. You can save the evaluated data to the MATLAB
workspace by selecting the Save to workspace
button.
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8. You can save the evaluated data to the MATLAB
workspace by selecting the Save to workspace
button.
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????? ?????? ??? ?- CF TBX

• ????? ?????? ???? ??
• - ???????? ??????? ?? CF TBX
• ???? ????. ???? ?????
• ????? ?? ??? ????? ????????
• ?????

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???????? ??????? ?? CF TBX

• The Curve Fitting Toolbox extends the MATLAB
  environment with integrated tools for
• Previewing and preprocessing data
• Creating, comparing, and managing models
• Using a variety of standard and custom fitting
  models
• Fitting with standard and robust methods
• Analyzing fits
• Nonparametric fitting using splines and
  interpolants
• Interpolation, extrapolation, differentiation,
  and integration of fits
• Introduction to Curve Fitting see here
• http//www.mathworks.com/products/curvefitting/dem
  os.jsp

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• 1. Open Curve Fitting Tool
• gtgtcftool

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2. Export Data for Fitting gtgtload census US
Population in 1790-1990
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3. Open Data GUI Click on Data button in CFTOOL
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• 4. Fitting the Data with the Fitting GUI.
• Click the Fitting button in the CV Tool window
• In Fitting window
• Fit Editor--gt New Fit.
• Select f. e. quadratic polynomial
• from the Polynomial list.
• Name the fit as poly2.
• Click the Apply button or select the
• Immediate apply check box.
• The library model, fitted coefficients,
• and goodness of fit statistics
• (sum of squares due to error - SSE,
• R-square and Root Mean Squared
• Error -RMSE)
• are displayed in the Results area.

Copy fit for saving current fitting
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• 5.To wiew Results with residuals
• In CF Tool window press View-gtResiduals-gtLine plot

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• 5.To exclude some points from the data
• In CF Tool window press Exclude and then in
  Exclude window
• Select data set -gt Check to exclude point and
  sign the points for exclusion

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• 5.To analyze the data
• In CF Tool window press Analysis and then in
  Analysis window sign the needed properties for
  analyzing and press Apply button

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• 5.To save your work
• In CF Tool window press File-gtGenerate
  M-file-gtFile Name-gtSave
• For run the saved file
• gtgt CFexample(cdata,pop) cdata,pop must be
  inputted before this command

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• Stages 1 3 from previous section.
• Open the Fitting window.
• Custom Equations -gt
• Center and scale data (for better fit) -gt
• New Equation

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• 2.In Create Custom Equation window choose General
  Equation
• In Equation enter needed equation, for example
• Yae-bxcedx
• In Equation name enter name

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????? ?? ??? DISTRIBUTION FITTING TOOL

• ????? ?????? ???? ??
• ????? ????????? ? STATISTICS TBX
• ??? dfittool

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????? ?? ??? DISTRIBUTION FITTING TOOL

• gtgtdata normrnd(.36, 1.4, 100, 1)
• gtgtdfittool

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????? ?? ??? DISTRIBUTION FITTING TOOL(????)
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????? ?? ??? DISTRIBUTION FITTING TOOL(????2)

• 4.

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????? ?? ??? DISTRIBUTION FITTING TOOL(????3)

• 4. Results