1. Exponential smoothing

The exponential smoothing model is a forecasting method based on the idea that past observations contain relevant information about the pattern of the time series.

Example:

Consider data on annual average temperatures in New York City during the years 1912 and 1971. Let’s fit a simple moving average model to this time series and forecast the temperature for the next 5 years.

Year Temperature
1912 49.9
1913 52.3
1914 49.4
1915 51.1
1916 49.4
1917 47.9
1918 49.8
1919 50.9
1920 49.3
1921 51.9
1922 50.8
1923 49.6
1924 49.3
1925 50.6
1926 48.4
1927 50.7
1928 50.9
1929 50.6
1930 51.5
1931 52.8
1932 51.8
1933 51.1
1934 49.8
1935 50.2
1936 50.4
1937 51.6
1938 51.8
1939 50.9
1940 48.8
1941 51.7
1942 51.0
1943 50.6
1944 51.7
1945 51.5
1946 52.1
1947 51.3
1948 51.0
1949 54.0
1950 51.4
1951 52.7
1952 53.1
1953 54.6
1954 52.0
1955 52.0
1956 50.9
1957 52.6
1958 50.2
1959 52.6
1960 51.6
1961 51.9
1962 50.5
1963 50.9
1964 51.7
1965 51.4
1966 51.7
1967 50.8
1968 51.9
1969 51.8
1970 51.9
1971 53.0

We will upload the data to the system.

We will carry out the analysis.

By clicking on Calculate we obtain the results.

The results are:

Accuracy Measurements

V1
MAPE 1.3452617
MAD 0.6865497
MSD 0.6562963

Average length
k
3

Estimation table

Data Fit Residuals
49.9
52.3
49.4 50.53333 -1.13333333
51.1 50.93333 0.16666667
49.4 49.96667 -0.56666667
47.9 49.46667 -1.56666667
49.8 49.03333 0.76666667
50.9 49.53333 1.36666667
49.3 50 -0.7
51.9 50.7 1.2
50.8 50.66667 0.13333333
49.6 50.76667 -1.16666667
49.3 49.9 -0.6
50.6 49.83333 0.76666667
48.4 49.43333 -1.03333333
50.7 49.9 0.8
50.9 50 0.9
50.6 50.73333 -0.13333333
51.5 51 0.5
52.8 51.63333 1.16666667
51.8 52.03333 -0.23333333
51.1 51.9 -0.8
49.8 50.9 -1.1
50.2 50.36667 -0.16666667
50.4 50.13333 0.26666667
51.6 50.73333 0.86666667
51.8 51.26667 0.53333333
50.9 51.43333 -0.53333333
48.8 50.5 -1.7
51.7 50.46667 1.23333333
51 50.5 0.5
50.6 51.1 -0.5
51.7 51.1 0.6
51.5 51.26667 0.23333333
52.1 51.76667 0.33333333
51.3 51.63333 -0.33333333
51 51.46667 -0.46666667
54 52.1 1.9
51.4 52.13333 -0.73333333
52.7 52.7 0
53.1 52.4 0.7
54.6 53.46667 1.13333333
52 53.23333 -1.23333333
52 52.86667 -0.86666667
50.9 51.63333 -0.73333333
52.6 51.83333 0.76666667
50.2 51.23333 -1.03333333
52.6 51.8 0.8
51.6 51.46667 0.13333333
51.9 52.03333 -0.13333333
50.5 51.33333 -0.83333333
50.9 51.1 -0.2
51.7 51.03333 0.66666667
51.4 51.33333 0.06666667
51.7 51.6 0.1
50.8 51.3 -0.5
51.9 51.46667 0.43333333
51.8 51.5 0.3
51.9 51.86667 0.03333333
53 52.23333 0.76666667

Forecasts

Lower limit of forecast Forecast Predicted upper limit
51.09756 51.86667 52.63578
51.46422 52.23333 53.00244
51.48645 52.25556 53.02466
51.59756 52.36667 53.13578
51.34941 52.11852 52.88763
51.51608 52.28519 53.05429