Time series

Model assessment

Tourist arrivals to NZ

Period is quarterly

Year Periiod No. in thousand1980 1 146.361

2 80.7823 85.4134 152.607

1981 5 144.6546 87.1967 91.2348 154.953

1982 9 139.19810 91.44611 96.10112 154.981

1983 13 145.1314 91.94315 104.77716 166.615

1984 17 155.10618 111.78819 112.02520 188.692

1985 21 184.4922 132.18623 131.9724 220.912

Draw graph of raw data

Tourists visiting NZ

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Number of tourists (000)

raw data

Write down the components of the

time series in context.

Identify and discuss any trend(s) in your series.

• The number of tourists arriving in NZ increases with time.

• NOT there is an increasing trend

Tourists visiting NZ

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Number of tourists (000)

raw data

Identify and describe at least 2 further features of your series. When discussing features you need to give

reasons that could account for them.

• There is a strong seasonal pattern shown where the no. of tourist arrival is highest during the 1st quarter for each year and lowest in the 2nd quarter.

• NOT there are seasonal effects

Tourists visiting NZ

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Number of tourists (000)

raw data

Check out where the highs and lows are by going back to the original data set.

Identify and describe at least 2 further features of your series. When discussing features you need to give

reasons that could account for them.

• Amplitude of seasonal variation is changing.(not constant) i.e. There are random fluctuations.

• NOT there are random effects

Tourists visiting NZ

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Number of tourists (000)

raw data

Identify and describe at least 2 further features of your series. When discussing features you need to give

reasons that could account for them.

• It seems that the data break into three sections follows:

• 1980 – 1987• 1987 – 1997• 1997 -• NOT there are ramps

or irregular components

Tourists visiting NZ

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Number of tourists (000)

raw data

Find out where changesin the pattern are

happening.

Find MM and CMM

Year Periiod No. in thousand MM CMM1980 1 146.361

2 80.782 116.33 85.413 115.9 116.14 152.607 117.5 116.7

1981 5 144.654 118.9 118.26 87.196 119.5 119.27 91.234 118.1 118.88 154.953 119.2 118.7

1982 9 139.198 120.4 119.810 91.446 120.4 120.411 96.101 121.9 121.212 154.981 122.0 122.0

1983 13 145.13 124.2 123.114 91.943 127.1 125.715 104.777 129.6 128.4

Add CMM to chartTourists visiting NZ

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Number of tourists (000)

raw data

CMM

Investigate modelsTourists visiting NZ

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Number of tourists (000)

raw data

CMM

Linear model

Tourists visiting NZ

y = 4.1861x + 74.022

R2 = 0.9787

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Number of tourists (000)

raw data

CMM

Linear (CMM)

OK - not perfect, but no one model

is likely to be perfect in these circumstances.

Quadratic model

Tourists visiting NZ

y = 0.0136x2 + 3.029x + 91.35

R2 = 0.9831

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Number of tourists (000)

raw data

CMM

Poly. (CMM)

Models overall quite well but doesn’t reflect the most

recent data very well.

Polynomial model-cubic

Tourists visiting NZ

y = -0.0004x 3 + 0.0638x2 + 1.2728x + 105.52

R2 = 0.9846

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period

Number of tourists (000)

raw data

CMM

Poly. (CMM)

Not really suitable as the end is

starting to curve downwards and

this does not reflect what is happening.

Polynomial model-quartic

Tourists visiting NZ

y = -5E-06x 4 + 0.0005x3 + 0.0133x2 + 2.2897x + 100.15

R2 = 0.9847

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period

Number of tourists (000)

raw data

CMM

Poly. (CMM)

Not really suitable as the end is

starting to curve downwards and

this does not reflect what is happening.

Polynomial model-5th power

Tourists visiting NZ

y = -7E-07x 5 + 0.0002x4 - 0.0116x 3 + 0.4152x2 - 3.1295x + 121.08

R2 = 0.9856

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Number of tourists (000)

raw data

CMM

Poly. (CMM)

Worse than the last two.

Polynomial model-6th power

Tourists visiting NZ

y = 9E-08x 6 - 2E-05x 5 + 0.0024x4 - 0.1185x 3 +

2.8464x2 - 27.142x + 193.82

R2 = 0.991

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Number of tourists (000)

raw data

CMM

Poly. (CMM)

Not good here but that doesn’t matter too much as we want to use the model for forecasting and so we

are more interested in how the modelreflects the most recent data

OK-may be rising too rapidly

Exponential

Tourists visiting NZ

y = 108.84e0.0178x

R2 = 0.976

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period

Number of tourists (000)

raw data

CMM

Expon. (CMM)

Good overall but not a very good model

at the end here.

Decide your model and state the trend in terms of the equation of the model

Tourists visiting NZ

y = 4.1861x + 74.022

R2 = 0.9787

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100

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Number of tourists (000)

raw data

CMM

Linear (CMM)

Tourists visiting NZ

y = 0.0136x2 + 3.029x + 91.35

R2 = 0.9831

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100

200

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Number of tourists (000)

raw data

CMM

Poly. (CMM)

Tourists visiting NZ

y = 9E-08x6 - 2E-05x 5 + 0.0024x4 - 0.1185x 3 +

2.8464x2 - 27.142x + 193.82

R2 = 0.991

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Number of tourists (000)

raw data

CMM

Poly. (CMM)

Tourists visiting NZ

y = 108.84e0.0178x

R2 = 0.976

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Number of tourists (000)

raw data

CMM

Expon. (CMM)

All four of these could be used.

It is not about one being better than the other. It is about you explaining why you made the choice and then evaluating your choice once you have finished the analysis.

Choosing on the basis of the R2 being higher is not recommended- all of these have high values and all are acceptable on this basis. R2 supports your choice.

Do you think it is additive or multiplicative?

Tourists visiting NZ

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Number of tourists (000)

raw data

CMM

Probably multiplicative as seasonal effects are increasing.

Tourists visiting NZ

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Number of tourists (000)

raw data

CMM

Choose your analysis

Linear additive Linear multiplicative

Quadratic additive Quadratic multiplicative

Linear - additive

€

y = 4.1861x + 74.022

Calculate trend, individual seasonal effects and averaged seasonal effects

Year Periiod No. in thousand MM CMM Trend ISE SE1980 1 146.361 78.2081 48.751075

2 80.782 116.3 82.3942 -61.67156253 85.413 115.9 116.1 86.5803 -30.7 -46.67109384 152.607 117.5 116.7 90.7664 35.9 58.9453313

1981 5 144.654 118.9 118.2 94.9525 26.5 48.7510756 87.196 119.5 119.2 99.1386 -32.0 -61.67156257 91.234 118.1 118.8 103.3247 -27.6 -46.67109388 154.953 119.2 118.7 107.5108 36.3 58.9453313

1982 9 139.198 120.4 119.8 111.6969 19.4 48.75107510 91.446 120.4 120.4 115.883 -29.0 -61.671562511 96.101 121.9 121.2 120.0691 -25.1 -46.671093812 154.981 122.0 122.0 124.2552 33.0 58.9453313

1983 13 145.13 124.2 123.1 128.4413 22.0 48.75107514 91.943 127.1 125.7 132.6274 -33.7 -61.671562515 104.777 129.6 128.4 136.8135 -23.6 -46.671093816 166.615 134.6 132.1 140.9996 34.5 58.9453313

1984 17 155.106 136.4 135.5 145.1857 19.6 48.75107518 111.788 141.9 139.1 149.3718 -27.4 -61.671562519 112.025 149.2 145.6 153.5579 -33.6 -46.671093820 188.692 154.3 151.8 157.744 36.9 58.9453313

Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 78.208 48.751 126.959 97.610

2 80.782 116.3 82.394 -61.672 20.723 142.4543 85.413 115.9 116.1 86.580 -30.664 -46.671 39.909 132.0844 152.607 117.5 116.7 90.766 35.941 58.945 149.712 93.662

1981 5 144.654 118.9 118.2 94.953 26.459 48.751 143.704 95.9036 87.196 119.5 119.2 99.139 -32.020 -61.672 37.467 148.8687 91.234 118.1 118.8 103.325 -27.593 -46.671 56.654 137.9058 154.953 119.2 118.7 107.511 36.277 58.945 166.456 96.008

1982 9 139.198 120.4 119.8 111.697 19.382 48.751 160.448 90.44710 91.446 120.4 120.4 115.883 -28.982 -61.672 54.211 153.11811 96.101 121.9 121.2 120.069 -25.072 -46.671 73.398 142.77212 154.981 122.0 122.0 124.255 33.004 58.945 183.201 96.036

1983 13 145.13 124.2 123.1 128.441 22.007 48.751 177.192 96.37914 91.943 127.1 125.7 132.627 -33.719 -61.672 70.956 153.61515 104.777 129.6 128.4 136.814 -23.586 -46.671 90.142 151.44816 166.615 134.6 132.1 141.000 34.524 58.945 199.945 107.670

1984 17 155.106 136.4 135.5 145.186 19.629 48.751 193.937 106.35518 111.788 141.9 139.1 149.372 -27.355 -61.672 87.700 173.46019 112.025 149.2 145.6 153.558 -33.551 -46.671 106.887 158.69620 188.692 154.3 151.8 157.744 36.894 58.945 216.689 129.747

1985 21 184.49 159.3 156.8 161.930 27.649 48.751 210.681 135.73922 132.186 167.4 163.4 166.116 -31.176 -61.672 104.445 193.85823 131.97 172.3 169.8 170.302 -37.859 -46.671 123.631 178.641

Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

Tourists visiting NZ

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period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Two things to notice..

• How well does the forecast reflect the situation? How reliable will it be to use this model?

• Identify SAV values that are higher or lower than the CMM.

Tourists visiting NZ

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Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your graphs. Print two graphs and hand them in.

Tourists visiting NZ

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period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Tourists visiting NZ

y = 4.1861x + 74.022

R2 = 0.9787

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100

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0 10 20 30 40 50 60 70 80 90

period

Number of tourists (000)

raw data

CMM

Linear (CMM)

Name your worksheet.Fit your data sheet to 1 page, print in

landscape and hand it in.Quarterly short-term tourist arrivals to NZ

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 78.208 48.751 126.959 97.610

2 80.782 116.3 82.394 -61.672 20.723 142.4543 85.413 115.9 116.1 86.580 -30.664 -46.671 39.909 132.0844 152.607 117.5 116.7 90.766 35.941 58.945 149.712 93.662

1981 5 144.654 118.9 118.2 94.953 26.459 48.751 143.704 95.9036 87.196 119.5 119.2 99.139 -32.020 -61.672 37.467 148.8687 91.234 118.1 118.8 103.325 -27.593 -46.671 56.654 137.9058 154.953 119.2 118.7 107.511 36.277 58.945 166.456 96.008

1982 9 139.198 120.4 119.8 111.697 19.382 48.751 160.448 90.44710 91.446 120.4 120.4 115.883 -28.982 -61.672 54.211 153.11811 96.101 121.9 121.2 120.069 -25.072 -46.671 73.398 142.77212 154.981 122.0 122.0 124.255 33.004 58.945 183.201 96.036

1983 13 145.13 124.2 123.1 128.441 22.007 48.751 177.192 96.37914 91.943 127.1 125.7 132.627 -33.719 -61.672 70.956 153.61515 104.777 129.6 128.4 136.814 -23.586 -46.671 90.142 151.44816 166.615 134.6 132.1 141.000 34.524 58.945 199.945 107.670

1984 17 155.106 136.4 135.5 145.186 19.629 48.751 193.937 106.35518 111.788 141.9 139.1 149.372 -27.355 -61.672 87.700 173.46019 112.025 149.2 145.6 153.558 -33.551 -46.671 106.887 158.69620 188.692 154.3 151.8 157.744 36.894 58.945 216.689 129.747

1985 21 184.49 159.3 156.8 161.930 27.649 48.751 210.681 135.73922 132.186 167.4 163.4 166.116 -31.176 -61.672 104.445 193.85823 131.97 172.3 169.8 170.302 -37.859 -46.671 123.631 178.64124 220.912 173.6 172.9 174.488 47.963 58.945 233.434 161.967

1986 25 204.005 176.2 174.9 178.675 29.083 48.751 227.426 155.25426 137.63 183.4 179.8 182.861 -42.156 -61.672 121.189 199.302

Drop your workbook into your teacher’s drop box.

Conclusions

Linear - multiplicative

€

y = 4.1861x + 74.022

Calculate trend, individual seasonal effects and averaged seasonal effects

Year Periiod No. in thousand MM CMM Trend ISE SE1980 1 146.361 78.208 1.185

2 80.782 116.3 82.394 0.7613 85.413 115.9 116.1 86.580 0.736 0.8054 152.607 117.5 116.7 90.766 1.308 1.246

1981 5 144.654 118.9 118.2 94.953 1.224 1.1856 87.196 119.5 119.2 99.139 0.731 0.7617 91.234 118.1 118.8 103.325 0.768 0.8058 154.953 119.2 118.7 107.511 1.306 1.246

1982 9 139.198 120.4 119.8 111.697 1.162 1.18510 91.446 120.4 120.4 115.883 0.759 0.76111 96.101 121.9 121.2 120.069 0.793 0.80512 154.981 122.0 122.0 124.255 1.271 1.246

1983 13 145.13 124.2 123.1 128.441 1.179 1.18514 91.943 127.1 125.7 132.627 0.732 0.76115 104.777 129.6 128.4 136.814 0.816 0.80516 166.615 134.6 132.1 141.000 1.261 1.246

1984 17 155.106 136.4 135.5 145.186 1.145 1.18518 111.788 141.9 139.1 149.372 0.803 0.76119 112.025 149.2 145.6 153.558 0.770 0.80520 188.692 154.3 151.8 157.744 1.243 1.246

1985 21 184.49 159.3 156.8 161.930 1.176 1.185

Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 78.208 1.185 92.709 123.469

2 80.782 116.3 82.394 0.761 62.704 106.1493 85.413 115.9 116.1 86.580 0.736 0.805 69.736 106.0454 152.607 117.5 116.7 90.766 1.308 1.246 113.126 122.444

1981 5 144.654 118.9 118.2 94.953 1.224 1.185 112.558 122.0296 87.196 119.5 119.2 99.139 0.731 0.761 75.447 114.5777 91.234 118.1 118.8 103.325 0.768 0.805 83.222 113.2728 154.953 119.2 118.7 107.511 1.306 1.246 133.996 124.326

1982 9 139.198 120.4 119.8 111.697 1.162 1.185 132.407 117.42610 91.446 120.4 120.4 115.883 0.759 0.761 88.190 120.16211 96.101 121.9 121.2 120.069 0.793 0.805 96.709 119.31412 154.981 122.0 122.0 124.255 1.271 1.246 154.865 124.348

1983 13 145.13 124.2 123.1 128.441 1.179 1.185 152.256 122.43014 91.943 127.1 125.7 132.627 0.732 0.761 100.933 120.81515 104.777 129.6 128.4 136.814 0.816 0.805 110.195 130.08616 166.615 134.6 132.1 141.000 1.261 1.246 175.734 133.683

1984 17 155.106 136.4 135.5 145.186 1.145 1.185 172.105 130.84618 111.788 141.9 139.1 149.372 0.803 0.761 113.676 146.89219 112.025 149.2 145.6 153.558 0.770 0.805 123.682 139.08520 188.692 154.3 151.8 157.744 1.243 1.246 196.603 151.396

1985 21 184.49 159.3 156.8 161.930 1.176 1.185 191.954 155.63422 132.186 167.4 163.4 166.116 0.809 0.761 126.418 173.69523 131.97 172.3 169.8 170.302 0.777 0.805 137.169 163.84824 220.912 173.6 172.9 174.488 1.277 1.246 217.473 177.248

Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

Tourists visiting NZ

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100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Two things to notice..

• How well does the forecast reflect the situation? How reliable will it be to use this model?

• Identify SAV values that are higher or lower than the CMM.

Tourists visiting NZ

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100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your graphs. Print two graphs and hand them in.

Tourists visiting NZ

y = 4.1861x + 74.022

R2 = 0.9787

0

100

200

300

400

500

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700

0 10 20 30 40 50 60 70 80 90

period

Number of tourists (000)

raw data

CMM

Linear (CMM)

Tourists visiting NZ

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100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your worksheet.Fit your data sheet to 1 page, print in

landscape and hand it in.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 78.208 1.185 92.709 123.469

2 80.782 116.3 82.394 0.761 62.704 106.1493 85.413 115.9 116.1 86.580 0.736 0.805 69.736 106.0454 152.607 117.5 116.7 90.766 1.308 1.246 113.126 122.444

1981 5 144.654 118.9 118.2 94.953 1.224 1.185 112.558 122.0296 87.196 119.5 119.2 99.139 0.731 0.761 75.447 114.5777 91.234 118.1 118.8 103.325 0.768 0.805 83.222 113.2728 154.953 119.2 118.7 107.511 1.306 1.246 133.996 124.326

1982 9 139.198 120.4 119.8 111.697 1.162 1.185 132.407 117.42610 91.446 120.4 120.4 115.883 0.759 0.761 88.190 120.16211 96.101 121.9 121.2 120.069 0.793 0.805 96.709 119.31412 154.981 122.0 122.0 124.255 1.271 1.246 154.865 124.348

1983 13 145.13 124.2 123.1 128.441 1.179 1.185 152.256 122.43014 91.943 127.1 125.7 132.627 0.732 0.761 100.933 120.81515 104.777 129.6 128.4 136.814 0.816 0.805 110.195 130.08616 166.615 134.6 132.1 141.000 1.261 1.246 175.734 133.683

1984 17 155.106 136.4 135.5 145.186 1.145 1.185 172.105 130.84618 111.788 141.9 139.1 149.372 0.803 0.761 113.676 146.89219 112.025 149.2 145.6 153.558 0.770 0.805 123.682 139.08520 188.692 154.3 151.8 157.744 1.243 1.246 196.603 151.396

1985 21 184.49 159.3 156.8 161.930 1.176 1.185 191.954 155.63422 132.186 167.4 163.4 166.116 0.809 0.761 126.418 173.69523 131.97 172.3 169.8 170.302 0.777 0.805 137.169 163.84824 220.912 173.6 172.9 174.488 1.277 1.246 217.473 177.248

Drop your workbook into your teacher’s drop box.

A linear model y = 4.1861x + 74.022 (y is the number of tourists and x represents the period)

is used for making the prediction since the R2 value is 0.9787 which is quite close to 1

indicating the CMM are closely clustered around the trend line.

Explain R2 in terms of the model (98% of the variability in the number of tourists is explained by the regression model y = 4.1861x + 74.022 )Also comment on the goodness of fit from your

forecast data. This is judged visually.

Conclusions

Goodness of fit

You will write down your forecast predictions making sure you have taken notice if figures

are in thousands.

Interpret the gradient (4.1861) of the linear model as ‘in each quarter numbers of tourists

are increasing by approximately 4200.

Forecast(s)

Calculation of seasonally adjusted data together with an interpretation of that data.

If the seasonally adjusted value is lower than the CMM, then the no. of tourist arriving to NZ for

that period is lower than expected. Give a specific example.

On the other hand if the seasonally adjusted value is higher than the CMM, then the no. of

tourist arriving to NZ during that period is more than expected.

Give a specific example.

A discussion of the relevance and usefulness of the forecast - An indication that conditions need to remain substantially the same

for the forecast to be valuable

Include a comment like - In order for the forecast to be accurate, we have to assume that the conditions for the tourist arrival to NZ need to remain substantially the same. That is no volcanic eruptions/ earthquakes occur in NZ that will decrease the number of tourist arriving.

Talk about how well you think your model performs.

A discussion of the relevance and usefulness of the forecast

The forecast estimates may be reasonably reliable for short times into the future when predicting the ‘low’ quarters, but for longer times, the estimate is liable to become less accurate. Give a comment on how the model shows this referring to your graph.

The forecasts are not reliable for the ‘high’ quarters as the model does not reflect the situation. This is evident in the over-plot of forecast values.

Potential sources of bias

The data collected are the actual numbers entering the country as tourists and hence there is not likely to be any bias.

Possible improvements to your model - additive

Obviously this model is not a good one for predictions as it does not reflect the most recent situation in the peak seasons.

An improvement would be to analyse the data using a multiplicative model as the seasonal effects are increasing in amplitude.

or

An improvement would be to just model (using an additive model) the most recent data but mention should be made that there appears to be a shift every 10 years and hence this model should be used with limitations.

Possible improvements to your model - multiplicative

By using a multiplicative model, you already have displayed an improvement but add.

This model is a reasonably good one for predictions in the short term however there may be a change in trend as indicated by the last 3 quarters of data compared to those forecast.

An improvement would be to just model (using an additive model) the most recent data but mention should be made that there appears to be a shift every 10 years and hence this model should be used with limitations.

Limitations of your analysis- linear additive

The model was quite accurate for the period 1986 to 1994 but is not a good one for forecasting beyond this time. This is obvious when comparing the forecast plot to the original data. The seasonally adjusted data also suggest that the model is more likely to be multiplicative as every value above 1994 is either above or below the centered moving mean and hence would be deemed unusual. This is unlikely to be the case.

Read pages 21 to 27 in your booklet.

‘Revision Notes and Procedure’

It will help you to understand exactly what is required and also helps with how you should word your answers. If you read these pages, you may not make careless errors.

Now compare all four models

Quadratic - additive

R2 = 0.9831

€

y = 0.0136x 2 + 3.029x + 91.35

Calculate trend, individual seasonal effects and averaged seasonal effects

Year Periiod No. in thousand MM CMM Trend ISE SE1980 1 146.361 94.393 48.751

2 80.782 116.3 97.462 -61.6723 85.413 115.9 116.1 100.559 -30.664 -46.6714 152.607 117.5 116.7 103.684 35.941 58.945

1981 5 144.654 118.9 118.2 106.835 26.459 48.7516 87.196 119.5 119.2 110.014 -32.020 -61.6727 91.234 118.1 118.8 113.219 -27.593 -46.6718 154.953 119.2 118.7 116.452 36.277 58.945

1982 9 139.198 120.4 119.8 119.713 19.382 48.75110 91.446 120.4 120.4 123.000 -28.982 -61.67211 96.101 121.9 121.2 126.315 -25.072 -46.67112 154.981 122.0 122.0 129.656 33.004 58.945

1983 13 145.13 124.2 123.1 133.025 22.007 48.75114 91.943 127.1 125.7 136.422 -33.719 -61.67215 104.777 129.6 128.4 139.845 -23.586 -46.67116 166.615 134.6 132.1 143.296 34.524 58.945

1984 17 155.106 136.4 135.5 146.773 19.629 48.75118 111.788 141.9 139.1 150.278 -27.355 -61.67219 112.025 149.2 145.6 153.811 -33.551 -46.67120 188.692 154.3 151.8 157.370 36.894 58.945

Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 94.393 48.751 143.144 97.610

2 80.782 116.3 97.462 -61.672 35.791 142.4543 85.413 115.9 116.1 100.559 -30.664 -46.671 53.888 132.0844 152.607 117.5 116.7 103.684 35.941 58.945 162.629 93.662

1981 5 144.654 118.9 118.2 106.835 26.459 48.751 155.586 95.9036 87.196 119.5 119.2 110.014 -32.020 -61.672 48.342 148.8687 91.234 118.1 118.8 113.219 -27.593 -46.671 66.548 137.9058 154.953 119.2 118.7 116.452 36.277 58.945 175.398 96.008

1982 9 139.198 120.4 119.8 119.713 19.382 48.751 168.464 90.44710 91.446 120.4 120.4 123.000 -28.982 -61.672 61.328 153.11811 96.101 121.9 121.2 126.315 -25.072 -46.671 79.644 142.77212 154.981 122.0 122.0 129.656 33.004 58.945 188.602 96.036

1983 13 145.13 124.2 123.1 133.025 22.007 48.751 181.776 96.37914 91.943 127.1 125.7 136.422 -33.719 -61.672 74.750 153.61515 104.777 129.6 128.4 139.845 -23.586 -46.671 93.174 151.44816 166.615 134.6 132.1 143.296 34.524 58.945 202.241 107.670

1984 17 155.106 136.4 135.5 146.773 19.629 48.751 195.524 106.35518 111.788 141.9 139.1 150.278 -27.355 -61.672 88.607 173.46019 112.025 149.2 145.6 153.811 -33.551 -46.671 107.140 158.69620 188.692 154.3 151.8 157.370 36.894 58.945 216.315 129.747

Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Two things to notice..

• How well does the forecast reflect the situation? How reliable will it be to use this model?

• Identify SAV values that are higher or lower than the CMM.

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your graphs. Print two graphs and hand them in.

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Tourists visiting NZ

y = 0.0136x2 + 3.029x + 91.35

R2 = 0.9831

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90

period

Number of tourists (000)

raw data

CMM

Poly. (CMM)

Name your worksheet.Fit your data sheet to 1 page, print in

landscape and hand it in.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 94.393 48.751 143.144 97.610

2 80.782 116.3 97.462 -61.672 35.791 142.4543 85.413 115.9 116.1 100.559 -30.664 -46.671 53.888 132.0844 152.607 117.5 116.7 103.684 35.941 58.945 162.629 93.662

1981 5 144.654 118.9 118.2 106.835 26.459 48.751 155.586 95.9036 87.196 119.5 119.2 110.014 -32.020 -61.672 48.342 148.8687 91.234 118.1 118.8 113.219 -27.593 -46.671 66.548 137.9058 154.953 119.2 118.7 116.452 36.277 58.945 175.398 96.008

1982 9 139.198 120.4 119.8 119.713 19.382 48.751 168.464 90.44710 91.446 120.4 120.4 123.000 -28.982 -61.672 61.328 153.11811 96.101 121.9 121.2 126.315 -25.072 -46.671 79.644 142.77212 154.981 122.0 122.0 129.656 33.004 58.945 188.602 96.036

1983 13 145.13 124.2 123.1 133.025 22.007 48.751 181.776 96.37914 91.943 127.1 125.7 136.422 -33.719 -61.672 74.750 153.61515 104.777 129.6 128.4 139.845 -23.586 -46.671 93.174 151.44816 166.615 134.6 132.1 143.296 34.524 58.945 202.241 107.670

1984 17 155.106 136.4 135.5 146.773 19.629 48.751 195.524 106.35518 111.788 141.9 139.1 150.278 -27.355 -61.672 88.607 173.46019 112.025 149.2 145.6 153.811 -33.551 -46.671 107.140 158.69620 188.692 154.3 151.8 157.370 36.894 58.945 216.315 129.747

Drop your workbook into your teacher’s drop box.

Read the conclusions for linearModels and adapt answers for

The quadratic model.

Quadratic - multiplicative

R2 = 0.9831

€

y = 0.0136x 2 + 3.029x + 91.35

Calculate trend, individual seasonal effects and averaged seasonal effects

Year Periiod No. in thousand MM CMM Trend ISE SE1980 1 146.361 94.393 1.185

2 80.782 116.3 97.462 0.7613 85.413 115.9 116.1 100.559 0.736 0.8054 152.607 117.5 116.7 103.684 1.308 1.246

1981 5 144.654 118.9 118.2 106.835 1.224 1.1856 87.196 119.5 119.2 110.014 0.731 0.7617 91.234 118.1 118.8 113.219 0.768 0.8058 154.953 119.2 118.7 116.452 1.306 1.246

1982 9 139.198 120.4 119.8 119.713 1.162 1.18510 91.446 120.4 120.4 123.000 0.759 0.76111 96.101 121.9 121.2 126.315 0.793 0.80512 154.981 122.0 122.0 129.656 1.271 1.246

1983 13 145.13 124.2 123.1 133.025 1.179 1.18514 91.943 127.1 125.7 136.422 0.732 0.76115 104.777 129.6 128.4 139.845 0.816 0.80516 166.615 134.6 132.1 143.296 1.261 1.246

1984 17 155.106 136.4 135.5 146.773 1.145 1.18518 111.788 141.9 139.1 150.278 0.803 0.76119 112.025 149.2 145.6 153.811 0.770 0.80520 188.692 154.3 151.8 157.370 1.243 1.246

1985 21 184.49 159.3 156.8 160.957 1.176 1.185

Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 94.393 1.185 111.894 123.469

2 80.782 116.3 97.462 0.761 74.171 106.1493 85.413 115.9 116.1 100.559 0.736 0.805 80.995 106.0454 152.607 117.5 116.7 103.684 1.308 1.246 129.225 122.444

1981 5 144.654 118.9 118.2 106.835 1.224 1.185 126.643 122.0296 87.196 119.5 119.2 110.014 0.731 0.761 83.723 114.5777 91.234 118.1 118.8 113.219 0.768 0.805 91.192 113.2728 154.953 119.2 118.7 116.452 1.306 1.246 145.140 124.326

1982 9 139.198 120.4 119.8 119.713 1.162 1.185 141.908 117.42610 91.446 120.4 120.4 123.000 0.759 0.761 93.606 120.16211 96.101 121.9 121.2 126.315 0.793 0.805 101.739 119.31412 154.981 122.0 122.0 129.656 1.271 1.246 161.597 124.348

1983 13 145.13 124.2 123.1 133.025 1.179 1.185 157.690 122.43014 91.943 127.1 125.7 136.422 0.732 0.761 103.820 120.81515 104.777 129.6 128.4 139.845 0.816 0.805 112.637 130.08616 166.615 134.6 132.1 143.296 1.261 1.246 178.596 133.683

1984 17 155.106 136.4 135.5 146.773 1.145 1.185 173.987 130.84618 111.788 141.9 139.1 150.278 0.803 0.761 114.365 146.89219 112.025 149.2 145.6 153.811 0.770 0.805 123.886 139.08520 188.692 154.3 151.8 157.370 1.243 1.246 196.137 151.396

1985 21 184.49 159.3 156.8 160.957 1.176 1.185 190.800 155.63422 132.186 167.4 163.4 164.570 0.809 0.761 125.242 173.695

Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Two things to notice..

• How well does the forecast reflect the situation? How reliable will it be to use this model?

• Identify SAV values that are higher or lower than the CMM.

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your graphs. Print two graphs and hand them in.

Tourists visiting NZ

y = 0.0136x2 + 3.029x + 91.35

R2 = 0.9831

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90

period

Number of tourists (000)

raw data

CMM

Poly. (CMM)

Tourists visiting NZ

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Name your worksheet.Fit your data sheet to 1 page, print in

landscape and hand it in.

Year Periiod No. in thousand MM CMM Trend ISE SE Forecast SAV1980 1 146.361 94.393 1.185 111.894 123.469

2 80.782 116.3 97.462 0.761 74.171 106.1493 85.413 115.9 116.1 100.559 0.736 0.805 80.995 106.0454 152.607 117.5 116.7 103.684 1.308 1.246 129.225 122.444

1981 5 144.654 118.9 118.2 106.835 1.224 1.185 126.643 122.0296 87.196 119.5 119.2 110.014 0.731 0.761 83.723 114.5777 91.234 118.1 118.8 113.219 0.768 0.805 91.192 113.2728 154.953 119.2 118.7 116.452 1.306 1.246 145.140 124.326

1982 9 139.198 120.4 119.8 119.713 1.162 1.185 141.908 117.42610 91.446 120.4 120.4 123.000 0.759 0.761 93.606 120.16211 96.101 121.9 121.2 126.315 0.793 0.805 101.739 119.31412 154.981 122.0 122.0 129.656 1.271 1.246 161.597 124.348

1983 13 145.13 124.2 123.1 133.025 1.179 1.185 157.690 122.43014 91.943 127.1 125.7 136.422 0.732 0.761 103.820 120.81515 104.777 129.6 128.4 139.845 0.816 0.805 112.637 130.08616 166.615 134.6 132.1 143.296 1.261 1.246 178.596 133.683

1984 17 155.106 136.4 135.5 146.773 1.145 1.185 173.987 130.84618 111.788 141.9 139.1 150.278 0.803 0.761 114.365 146.89219 112.025 149.2 145.6 153.811 0.770 0.805 123.886 139.08520 188.692 154.3 151.8 157.370 1.243 1.246 196.137 151.396

1985 21 184.49 159.3 156.8 160.957 1.176 1.185 190.800 155.63422 132.186 167.4 163.4 164.570 0.809 0.761 125.242 173.695

Drop your workbook into your teacher’s drop box.

Read these conclusion and adapt for the

quadratic multiplicative model.

Compare all four modelsTourists visiting NZ

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period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Tourists visiting NZ

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period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Tourists visiting NZ

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0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Tourists visiting NZ

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0 10 20 30 40 50 60 70 80 90 100

period

Number of tourists (000)

raw data

CMM

Forecast

SAV

Using only the last few years’ data from 1997 - additive model

No. of tourists visiting NZ since 1997

y = 6.7878x + 333.95

R2 = 0.9162y = 0.6384x2 - 4.0654x + 372.47

R2 = 0.9919

200

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550

600

0 2 4 6 8 10 12 14 16 18

Period (Quarterly)

Number of tourists

raw data

CMM

Linear (CMM)

Poly. (CMM)

Use quadratic model

€

y = 0.6384x 2 − 4.0654 + 372.47

R2 = 0.9912

Forecasts and SAV

No. of tourists visiting NZ since 1997

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0 5 10 15 20 25

Period (Quarterly)

Number of tourists

raw data

CMM

Forecast

SAV

This is probably the best model to forecast from.

No. of tourists visiting NZ since 1997

200

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0 5 10 15 20 25

Period (Quarterly)

Number of tourists

raw data

CMM

Forecast

SAV