Curve Fitting2
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Curve Fitting2 Curve Fitting2 Presentation Transcript

  • Method Of Moving averages Secular Trend Time series
  • The Method of least Squares Curve fitting
    •  
    • The Technique of obtaining the mathematical curve by the principle of least squares is known as curve fitting
    •  
  • Two Types of trends are there
    • Linear Trend is one which gives the straight line when Plotted the values on a graph paper
    • If we are getting non-linear curves like parabola,exponential curve,logistic curve We can say it is a non-linear Trend.
  • We can fit the curve by Principle of least Squares
    • “The sum of the squares of the deviations between the observed values and the trend values is least”
  • Lines of Best Fit
    • If we are drawing a straight line by the method of curve fitting based on the principle of least squares ,such line we can say line of best fit
  • Equation of least squares
    • y = a + bx where y is the value for dependent Variable
    • a and b are constants
    • For finding a and b we will apply the method of least Squares
  • How can we find the value of a and b
    • Let E be the sum of the squares of deviations of all the original values.
    • E=( y – (a + bx))²
    • By the method of calculus ,partial differential coefficients of E with respect to a and b are zero.Thus we will get two normal Equations
    •  y=na +b  x
    •  xy= a  x +b  x²
    • Solve these two Equations we will get a and b
    • Substituting these values in the equation y=a+bx we get the trend Equation Substitute the various values in the trend Equation
  • Question No 1
    • The following are the annual Profits in 1000s of Rs in a textiles Business
    • Year 1998 1999 2000 2001 2002 2003 2004
    Profit in 60 72 75 65 80 85 95 Thousands a.Using the method of least squares fit a straight line. b.Also Make an estimate profit in 2005 c.Estimate the trend Values for all the years? d.Compute short term Fluctuations? e.Plot the given values and trend values on a graph?
  • Answer
    • a=532/7=76 b=136/28
    • The Trend Eqn is y=76 + 4.86xIn the year 2005 x=4 then y=76+(4.86*4)=95.44
    • Therefore the profit for the year 2005 will be 95.44 thousand Rs.
    28 136 0 532 9 4 1 0 1 4 9 -180 -144 -75 0 80 170 285 -3 -2 -1 0 1 2 3 60 72 75 65 80 85 95 1998 1999 2000 2001 2002 2003 2004 x² xy Deviation from Mid Year1964 X=t-1964 Profit(ooo’s Rs) (y) Year(t)
  • Contd…….
    • The Equation for St. line Y=a+bx
    • The Normal Eqns are
    •  y= na +b  x
    •  xy= a  x +b  x²
    • If  x= 0 then the eqn become
    • a =  y / n and b=  xy /  x²
  • Contd…….c
    • Trend Values
    • If x=-3 y=76+(4.86*-3)=61.42
    • If x=-2 y=76+(4.86*-2)=66.28
    • If x=-1 y=76+(4.86*-1)=71.14
    • If x = 0 y=76+(4.86*-0)=76
    • If x= 1 y=76+(4.86*-1)=80.86
    • If x = 2 y=76+(4.86*-2)=85.72
    • If x = 3 y=76+(4.86*3)=90.58
  • Contd…….d
    • Short Time Fluctuations=Actual Value-Trend Values(y-y)
    • Year Actual Value Y Trend ValuesY Short Time
    • Fluctuations
  • Equations
    • From The eqn
    • Y= a+bx
    •  y = na + b  x
    •  xy = a  x + b  x ²
    • Solving the two equations we will get a and b.
    • Substitute the values in the eqn Y= a+bx,we get the trend equation
    • The trend values can be obtained by substituting the various values of x in the trend equation
  • Home work 1
    • Fit a Straight line trend to the following series by the method of least squares
    • Year Production in steel(‘000 tones )
    • 1994 10
      • 1995 13
      • 1996 12
      • 1997 14
      • 1998 12
      • 1999 16
      • 2000 14
  • Home work 2
    • The following are the annual profits in thousands of rupees in a certain business
    • Year Production in steel(‘000 tones )
    • 1994 60
      • 1995 72
      • 1996 75
      • 1997 65
      • 1998 80
      • 1999 85
      • 2000 95
      • a.Using the method of least squares fit a straight line to the above data .
      • b. Also make an estimate of profit in 2001
      • c.Estimate the trend values for all the years.
      • d. Compute short term fluctuations
      • e. Plot the given values and trend values on a given graph
  • Home work 3
    • Below are given the figures of production in thousand tones of a sugar factory
    • Year Production in steel(‘000 tones )
    • 1994 60
      • 1995 72
      • 1996 75
      • 1997 65
      • 1998 80
      • 1999 85
      • 2000 95
      • Fit a straight line by the method of least Squares and find the trend values
      • b. What is the monthly increase in production
      • C.Eliminate the trend by assuming additive model and multiplicative model
  • Question 2
    • Compute the trend values by the method of least squares from the data given below
    • 1992 56
      • 1993 55
      • 1994 51
      • 1995 47
      • 1996 42
      • 1997 38
      • 1998 35
      • 1999 32
    • When there are two middle years , take average of them as assumed average and take deviation from it . Multiplying by 2 to avoid decimals
    • Take deviations from 1965.5 and multiply by 2