Fundamental Math

IT100 Session 8: Trends and seasonal effects
5 Sept 2017

Agenda

Agenda

  1. Introduction to Forecasting
  2. Linear, exponential, logrithmic trends
  3. Seasonal effects
  4. Measuring Accuracy of Forecasting

1. Introduction to Forecasting

Forecasting

  • Uses mathematical models:

    • to determine general trends in historical data
    • to project possible futures based on trends discovered
    • to monitor variations from expected trends
    • uses statistics to determine the level of confidence

  • Assumes that the content does not change

    • Subject to changes in technology, environment, politics

Drug Pharmacokinetics: Indomethacin

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Orange Tree Growth

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2. Determining trends

Regression

\[ b = \bar y - m \bar x \] \[ m = { \sum_{i=1}^n (x_i -\bar x)(y_i - \bar y)\over \sum_{i=1}^n (x_i -\bar x)^2 } \]

Excel Trendlines

Population Modelling

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Challenge

Use the Down Jones Industrial Average

  • Determine the best trendline for this dataset
  • Use the trendline to determine to recalculate the dataset
  • Determine the difference between the forecast
  • Use the trendline to determine the subsequent 6 periods

3. Seasonal effects

Air Passengers

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Trend

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Steps of Seasonal models

  1. Determine the annual totals
  2. Establish the year to year trends
  3. Calculate a projections to the next 5 years
  4. Determine the fraction of the year for each season
  5. Calculate the average seasonal fraction for the dataset
  6. Apply the seasonal fraction to the annual projection
  7. Recalculate the seasonal data based on annual projects.

UK Gas Consumption Data

\[ \begin{array}{lrrrrr} Year & Qtr1 & Qtr2 & Qtr3 & Qtr4 & Total\\ 1972 & 317.0 & 230.5 & 152.1 & 336.2 & 1035.8\\ 1973 & 371.4 & 240.1 & 158.5 & 355.4 & 1125.4\\ 1974 & 449.9 & 286.6 & 179.3 & 403.4 & 1319.2\\ 1975 & 491.5 & 321.8 & 177.7 & 409.8 & 1400.8\\ 1976 & 593.9 & 329.8 & 176.1 & 483.5 & 1583.3\\ 1977 & 584.3 & 395.4 & 187.3 & 485.1 & 1652.1\\ 1978 & 669.2 & 421.0 & 216.1 & 509.1 & 1815.4\\ 1979 & 827.7 & 467.5 & 209.7 & 542.7 & 2047.6\\ 1980 & 840.5 & 414.6 & 217.7 & 670.8 & 2143.6\\ 1981 & 848.5 & 437.0 & 209.7 & 701.2 & 2196.4\\ 1982 & 925.3 & 443.4 & 214.5 & 683.6 & 2266.8\\ 1983 & 917.3 & 515.5 & 224.1 & 694.8 & 2351.7\\ 1984 & 989.4 & 477.1 & 233.7 & 730.0 & 2430.2\\ 1985 & 1087.0 & 534.7 & 281.8 & 787.6& 2691.1\\ 1986 & 1163.9 & 613.1 & 347.4 & 782.8& 2907.2\\ \end{array} \]

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Stage

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Seasonal effects as a fraction

\[ \begin{array}{lrrrr} Yr & Q1 & Q2 & Q3 & Q4\\ 1972 & 0.306043638 & 0.222533308 & 0.146843020 & 0.324580035\\ 1973 & 0.330015994 & 0.213346366 & 0.140838813 & 0.315798827\\ 1974 & 0.341040024 & 0.217252881 & 0.135915706 & 0.305791389\\ 1975 & 0.350870931 & 0.229725871 & 0.126856082 & 0.292547116\\ 1976 & 0.375102634 & 0.208299122 & 0.111223394 & 0.305374850\\ 1977 & 0.353671085 & 0.239331760 & 0.113370861 & 0.293626294\\ 1978 & 0.368623995 & 0.231904814 & 0.119037127 & 0.280434064\\ 1979 & 0.404229342 & 0.228316077 & 0.102412581 & 0.265042000\\ 1980 & 0.392097406 & 0.193412950 & 0.101558127 & 0.312931517\\ 1981 & 0.386313968 & 0.198961938 & 0.095474413 & 0.319249681\\ 1982 & 0.408196577 & 0.195606141 & 0.094626787 & 0.301570496\\ 1983 & 0.390058256 & 0.219203130 & 0.095292767 & 0.295445848\\ 1984 & 0.407126985 & 0.196321290 & 0.096164925 & 0.300386799\\ 1985 & 0.403924046 & 0.198691985 & 0.104715544 & 0.292668426\\ 1986 & 0.400350853 & 0.210890204 & 0.119496423 & 0.269262521\\ Mean & 0.374511049 & 0.213586522 & 0.113588438 & 0.298313991\\ \end{array} \]

Seasonal effects

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Applying seasonal effects to the trend

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4. Measuring Accuracy of Forecasting

Measuring goodness of fit

Correlation coefficent

  • A measure of the linear relationship
  • 1 if Positive correlation
  • -1 if Negative correlation

\( \large R^2 \)

  • Determine if the error pattern follows the curve
  • Suggests the fraction of the behavior explained by the model

Analysis of Variance

  • A measure of the trends in the pattern of errors
  • Determines if the model explains the pattern

MEAN ABSOLUTE DEVIATION

\[ \huge MAD = \sum_{i=1}^n {|obs_i - est_i| \over n} \]

  • observed value
  • estimated value
  • number of measurements
Year Obs Predict Abs Dev
1972 1035.8 1048.109 12.309167
1973 1125.4 1174.252 48.851667
1974 1319.2 1300.394 18.805833
1975 1400.8 1426.537 25.736667
1976 1583.3 1552.679 30.620833
1977 1652.1 1678.822 26.721667
1978 1815.4 1804.964 10.435833
1979 2047.6 1931.107 116.493333
1980 2143.6 2057.249 86.350833
1981 2196.4 2183.392 13.008333
1982 2266.8 2309.534 42.734167
1983 2351.7 2435.677 83.976667
1984 2430.2 2561.819 131.619167
1985 2691.1 2687.962 3.138333
1986 2907.2 2814.104 93.095833

Growth of Chicks

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Model \( \large R^2 \) \( MAD \)
Linear Model: 0.9855 5.106
Exponental Model: 0.9899 5.074

Challenge

Use the Air Passenger Data

  • Determine the annual number of passengers
  • Use the annual figures to determine the best trendline
  • Use the trendline to determine the MAD and the next 5 years
  • Determine the seasonal effects
  • Apply the seasonal adjustment and determine the new MAD
  • Apply the seasonal adjustments to the forecast