Experiments with Fuzzy Methods for Forecasting Time Series as Alternatives to Classical Methods
Abstract
:1. Introduction
2. Data and Methods
- The volume index of production in construction (2015 = 100), seasonally and calendar-adjusted, for 20 European countries;
- The volume index of production in the industry for electricity, gas, steam and air conditioning supply (2015 = 100), seasonally and calendar-adjusted, for 29 European countries;
- The index of deflated turnover in retail trade, except for motor vehicles and motorcycles (2015 = 100), seasonally and calendar-adjusted, for 28 European countries.
- Set the universe of discourse
- Divide the universe of discourse into several intervals
- Fuzzify the datasets
- Setting the fuzzy logical relationship and group
- Defuzzification.
- Define U as the universe of discourse starting from the range of time series data, setting , with D1 and D2 being two convenient positive numbers
- Divide U into equal length intervals u1, u2, … uk. The number of intervals should be equal to the number of linguistic variables A1, A2, … Ak
- Build the fuzzy sets Ai according to the previously set intervals and apply triangular membership rule to each one
- Fuzzify the available data and set the fuzzy logical relations where Ai corresponds to the fuzzified value at time step n and Aj to the time step n + 1.
- Define the universe of discourse (U) starting from the range of time series data;
- Divide the set U into equal-length intervals
- Determine the respective values of linguistic variable
- Fuzzify the input data
- Select the parameter W > 1, and compute the fuzzy relationships matrix
- Defuzzify the obtained results.
3. Results
4. Conclusions and Future Work
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Country | Min | 1sq QU | Median | Mean | 3rd QU | Max | VAR | SD | SE | Order of Integration |
---|---|---|---|---|---|---|---|---|---|---|
Belgium | 96.4 | 100.62 | 101.8 | 101.95 | 103 | 107.3 | 3.41 | 1.84 | 0.29 | 1 |
Bulgaria | 73.7 | 89.02 | 89.8 | 89.55 | 90.82 | 95.4 | 11.68 | 3.41 | 0.55 | 1 |
Czechia | 90.4 | 99.3 | 106.9 | 104.35 | 109.4 | 112.6 | 37.59 | 6.13 | 0.99 | 1 |
Denmark | 103.6 | 109.97 | 112.9 | 112.49 | 114.62 | 125.9 | 14.47 | 3.8 | 0.61 | 1 |
Germany | 101.4 | 109.25 | 109.9 | 110.63 | 112.35 | 119.1 | 10.38 | 3.22 | 0.52 | 1 |
Spain | 95.7 | 102.32 | 104.7 | 103.73 | 105.6 | 110 | 9.84 | 3.13 | 0.5 | 1 |
France | 98.4 | 101.25 | 102.7 | 102.65 | 103.55 | 107.1 | 4.1 | 2.02 | 0.32 | 2 |
Croatia | 98.5 | 106.47 | 112.75 | 113.19 | 118.77 | 131.4 | 59.73 | 7.72 | 1.25 | 1 |
Italy | 95.5 | 101.8 | 102.7 | 103.13 | 104.97 | 112.1 | 9.59 | 3.09 | 0.5 | 0 |
Luxembourg | 94.6 | 105.4 | 108.45 | 108.38 | 111.25 | 125.4 | 33.24 | 5.76 | 0.93 | 0 |
Hungary | 87.8 | 109.82 | 132.5 | 129.62 | 150.3 | 167.1 | 584.79 | 24.18 | 3.92 | 1 |
Netherlands | 112.2 | 118.42 | 124 | 123.18 | 127.55 | 132.6 | 32.42 | 5.69 | 0.92 | 1 |
Austria | 101 | 110.07 | 116.85 | 116.45 | 122.25 | 127.9 | 51.63 | 7.18 | 1.16 | 1 |
Poland | 85.1 | 101.5 | 117.65 | 112.87 | 122.65 | 129.6 | 153.12 | 12.37 | 2 | 1 |
Portugal | 96.6 | 99.37 | 101.45 | 101.19 | 103.3 | 106.6 | 7.12 | 2.66 | 0.43 | 0 |
Romania | 82.9 | 88.7 | 94.1 | 99.12 | 110.82 | 135.7 | 187.49 | 13.69 | 2.22 | 2 |
Slovenia | 78.3 | 109.5 | 126.55 | 122.38 | 134.42 | 144.1 | 240.89 | 15.52 | 2.51 | 1 |
Slovakia | 87.8 | 93.12 | 96.35 | 96.28 | 99.2 | 104.9 | 19.69 | 4.43 | 0.71 | 2 |
Finland | 109.2 | 111.22 | 113.1 | 112.95 | 114.45 | 118.1 | 4.37 | 2.09 | 0.33 | 1 |
Sweden | 105 | 116.45 | 121.6 | 119.68 | 122.77 | 129.4 | 25.35 | 5.03 | 0.81 | 1 |
Country | Min | 1sq QU | Median | Mean | 3rd QU | Max | VAR | SD | SE | Order of Integration |
---|---|---|---|---|---|---|---|---|---|---|
Belgium | 92.6 | 110.35 | 121.65 | 122.2 | 132.4 | 159.9 | 219.59 | 14.81 | 2.4 | 2 |
Bulgaria | 75.2 | 82.32 | 85.7 | 87.2 | 92.67 | 101.4 | 50.36 | 7.09 | 1.15 | 2 |
Czechia | 93.1 | 102.02 | 106.8 | 106.27 | 111.6 | 115.8 | 37.19 | 6.09 | 0.98 | 1 |
Denmark | 81.9 | 96.12 | 102.55 | 102.68 | 108.6 | 123.3 | 67.67 | 8.22 | 1.33 | 0 |
Germany | 88.7 | 93.55 | 98.35 | 97.24 | 100.8 | 106.3 | 20.83 | 4.56 | 0.74 | 1 |
Estonia | 64.7 | 89.55 | 115.95 | 107.56 | 121.15 | 149 | 573.3 | 23.94 | 3.88 | 2 |
Ireland | 104 | 112.85 | 115.5 | 117.23 | 121.87 | 132.9 | 47.79 | 6.91 | 1.12 | 0 |
Greece | 88.3 | 101.02 | 106.65 | 105.8 | 110.45 | 131.4 | 67.52 | 8.21 | 1.33 | 1 |
Spain | 88.4 | 94.6 | 97.35 | 97.43 | 99.17 | 107.5 | 20.54 | 4.53 | 0.73 | 1 |
France | 94.9 | 97.85 | 99.55 | 100.01 | 101.1 | 110.8 | 11.68 | 3.41 | 0.55 | 1 |
Croatia | 90.1 | 104.52 | 110.55 | 110.34 | 115.92 | 121.9 | 54.83 | 7.4 | 1.2 | 1 |
Italy | 91.6 | 101.55 | 102.95 | 103.11 | 105.27 | 111.4 | 14.4 | 3.79 | 0.61 | 1 |
Cyprus | 103.4 | 108.6 | 110.85 | 111.25 | 112.95 | 120.8 | 15.9 | 3.98 | 0.64 | 0 |
Latvia | 93 | 101.9 | 111.4 | 110.37 | 117.95 | 128.4 | 86.67 | 9.3 | 1.51 | 1 |
Lithuania | 93 | 98.97 | 102.8 | 102.48 | 105.82 | 116.7 | 31.44 | 5.6 | 0.9 | 0 |
Luxembourg | 81.4 | 90.92 | 96.25 | 97.29 | 104.42 | 109.4 | 63.53 | 7.97 | 1.29 | 2 |
Hungary | 93.3 | 96.4 | 99.35 | 99.26 | 101.05 | 116.2 | 14.79 | 3.84 | 0.62 | 1 |
Malta | 74.2 | 128.52 | 150.7 | 144.28 | 162.67 | 208.7 | 1012.44 | 31.81 | 5.16 | 1 |
Netherlands | 102.8 | 105.72 | 110 | 112.86 | 119.62 | 133 | 77.63 | 8.81 | 1.42 | 2 |
Austria | 104 | 112.15 | 119.45 | 118.47 | 124.87 | 133.2 | 69.5 | 8.33 | 1.35 | 2 |
Poland | 91.7 | 97.32 | 106.9 | 105.45 | 111.07 | 120.3 | 54.89 | 7.4 | 1.2 | 1 |
Portugal | 92.2 | 111.12 | 116.55 | 118.68 | 126.2 | 146.4 | 141.07 | 11.87 | 1.92 | 1 |
Romania | 93.9 | 100.97 | 102.95 | 102.62 | 105.32 | 110.5 | 15.58 | 3.94 | 0.64 | 1 |
Slovenia | 86.5 | 95.75 | 99 | 98.77 | 101.02 | 113.5 | 23.83 | 4.88 | 0.79 | 1 |
Slovakia | 100.7 | 106 | 108.7 | 108.56 | 112.07 | 117 | 14.02 | 3.74 | 0.6 | 2 |
Finland | 93.2 | 96.35 | 99 | 99.15 | 100.72 | 110.8 | 14.51 | 3.8 | 0.61 | 1 |
Sweden | 91.6 | 99.8 | 101.2 | 101.56 | 103.65 | 114.5 | 14.08 | 3.75 | 0.6 | 1 |
Norway | 86.8 | 94.15 | 97.95 | 98.44 | 103 | 112.5 | 37.45 | 6.12 | 0.99 | 0 |
Switzerland | 79.9 | 95.32 | 100.8 | 101.37 | 107.07 | 121 | 61.67 | 7.85 | 1.27 | 1 |
Country | Min | 1sq QU | Median | Mean | 3rd QU | Max | VAR | SD | SE | Order of Integration |
---|---|---|---|---|---|---|---|---|---|---|
Belgium | 98.4 | 99.82 | 100.3 | 100.42 | 100.77 | 102.6 | 0.82 | 0.9 | 0.14 | 1 |
Bulgaria | 109.6 | 112.72 | 115.9 | 116.5 | 120.75 | 128.8 | 21.38 | 4.62 | 0.75 | 2 |
Czechia | 106.9 | 113.45 | 116.8 | 116.99 | 121.75 | 125.2 | 26.74 | 5.17 | 0.83 | 1 |
Denmark | 101.3 | 102.5 | 103.55 | 103.48 | 104.6 | 105.3 | 1.59 | 1.26 | 0.2 | 1 |
Germany | 103.1 | 106.15 | 107.95 | 108.29 | 110.8 | 112.7 | 6.91 | 2.62 | 0.42 | 1 |
Estonia | 103.8 | 105.92 | 107.5 | 108.53 | 110.75 | 115.8 | 10.58 | 3.25 | 0.52 | 1 |
Ireland | 107 | 111.85 | 116.35 | 115.83 | 120.27 | 123.8 | 22.26 | 4.71 | 0.76 | 0 |
Greece | 97.1 | 100.62 | 102.2 | 101.96 | 103.25 | 106.1 | 4.12 | 2.03 | 0.32 | 1 |
Spain | 102.6 | 105.22 | 105.8 | 106.21 | 107.37 | 109.7 | 2.87 | 1.69 | 0.27 | 1 |
France | 105.1 | 107.9 | 110.1 | 109.98 | 112.4 | 114.3 | 8.12 | 2.85 | 0.46 | 1 |
Croatia | 102 | 106.07 | 111.75 | 110.06 | 114.12 | 116.8 | 20.76 | 4.55 | 0.73 | 1 |
Cyprus | 106.1 | 113.5 | 117.25 | 117.01 | 120.45 | 125.1 | 24.28 | 4.92 | 0.79 | 1 |
Latvia | 103.6 | 108.2 | 111.4 | 110.61 | 112.95 | 119.7 | 13.3 | 3.64 | 0.59 | 1 |
Lithuania | 109.2 | 112.4 | 119.8 | 118.98 | 124.15 | 131.2 | 41.51 | 6.44 | 1.04 | 1 |
Luxembourg | 28.2 | 30.3 | 30.9 | 31.06 | 32.2 | 33.2 | 1.52 | 1.23 | 0.2 | 1 |
Hungary | 107 | 112.87 | 118.75 | 118.75 | 124.5 | 132.2 | 50.62 | 7.11 | 1.15 | 1 |
Malta | 104.2 | 108.42 | 110.35 | 112.26 | 117 | 123.1 | 24.77 | 4.97 | 0.8 | 1 |
Netherlands | 102.5 | 105.32 | 107.95 | 107.64 | 109.85 | 112.5 | 6.46 | 2.54 | 0.41 | 1 |
Austria | 100.7 | 102.22 | 103.1 | 102.93 | 103.67 | 106.2 | 1.51 | 1.23 | 0.19 | 1 |
Poland | 109.6 | 114.77 | 119.35 | 119.6 | 124.3 | 132 | 39.71 | 6.3 | 1.02 | 1 |
Portugal | 103.7 | 108.45 | 111.7 | 112.09 | 115.52 | 124.5 | 22.65 | 4.75 | 0.77 | 2 |
Romania | 117.3 | 128.5 | 132.85 | 133.76 | 139.45 | 152.2 | 73.99 | 8.6 | 1.39 | 1 |
Slovenia | 110.8 | 112.82 | 116.15 | 117.53 | 121.45 | 129.1 | 28.21 | 5.31 | 0.86 | 2 |
Slovakia | 105.4 | 109.52 | 110.2 | 110.75 | 112.42 | 115.9 | 6.67 | 2.58 | 0.41 | 3 |
Finland | 103.2 | 105.27 | 106.5 | 107.06 | 109.2 | 111.9 | 5.53 | 2.35 | 0.38 | 2 |
Sweden | 102.4 | 104.37 | 105.75 | 105.93 | 107.55 | 110.9 | 4.13 | 2.03 | 0.32 | 0 |
Norway | 100.2 | 101.9 | 102.4 | 102.42 | 102.9 | 105.2 | 0.84 | 0.91 | 0.14 | 0 |
Switzerland | 98.3 | 100.8 | 101.5 | 101.39 | 101.87 | 103.4 | 1.35 | 1.16 | 0.18 | 1 |
Accuracy Indicator | Fuzzy Time Series Forecasting Method | Econometric Method | The Volume Index of Production in Construction (20 Time Series) | The Volume Index of Production in Industry for Electricity, Gas, Steam and Air Conditioning Supply (29 Time Series) | The Index of Deflated Turnover in Retail Trade, Except for Motor Vehicles and Motorcycles (28 Time Series) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
h = 2 | h = 4 | h = 6 | h = 2 | h = 4 | h = 6 | h = 2 | h = 4 | h = 6 | |||
MAE | Abbasov–Mamedova | SES | 8 | 11 | 7 | 14 | 13 | 6 | 10 | 12 | 11 |
Abbasov–Mamedova | Holt | 7 | 10 | 9 | 15 | 14 | 7 | 7 | 8 | 6 | |
Abbasov–Mamedova | ARIMA | 7 | 8 | 7 | 14 | 15 | 7 | 6 | 7 | 8 | |
NFTS | SES | 8 | 10 | 7 | 13 | 12 | 6 | 10 | 8 | 11 | |
NFTS | Holt | 8 | 9 | 9 | 14 | 13 | 9 | 6 | 10 | 4 | |
NFTS | ARIMA | 8 | 8 | 7 | 14 | 12 | 9 | 4 | 12 | 7 | |
MAPE | Abbasov–Mamedova | SES | 8 | 10 | 7 | 14 | 13 | 6 | 10 | 12 | 12 |
Abbasov–Mamedova | Holt | 7 | 9 | 9 | 15 | 12 | 7 | 7 | 8 | 6 | |
Abbasov–Mamedova | ARIMA | 7 | 8 | 7 | 15 | 13 | 7 | 6 | 7 | 8 | |
NFTS | SES | 9 | 10 | 8 | 13 | 12 | 6 | 10 | 13 | 11 | |
NFTS | Holt | 8 | 9 | 9 | 14 | 13 | 9 | 6 | 10 | 4 | |
NFTS | ARIMA | 8 | 8 | 7 | 14 | 11 | 8 | 4 | 8 | 7 | |
RMSE | Abbasov–Mamedova | SES | 8 | 11 | 6 | 13 | 12 | 6 | 9 | 12 | 12 |
Abbasov–Mamedova | Holt | 7 | 10 | 9 | 14 | 14 | 6 | 6 | 8 | 5 | |
Abbasov–Mamedova | ARIMA | 7 | 8 | 7 | 14 | 13 | 6 | 6 | 7 | 7 | |
NFTS | SES | 9 | 10 | 8 | 16 | 13 | 7 | 8 | 13 | 11 | |
NFTS | Holt | 8 | 11 | 9 | 13 | 14 | 8 | 6 | 9 | 3 | |
NFTS | ARIMA | 7 | 10 | 7 | 16 | 13 | 9 | 5 | 8 | 6 | |
MASE | Abbasov–Mamedova | SES | 8 | 11 | 7 | 14 | 13 | 6 | 10 | 12 | 11 |
Abbasov–Mamedova | Holt | 7 | 10 | 9 | 15 | 14 | 7 | 7 | 10 | 6 | |
Abbasov–Mamedova | ARIMA | 7 | 8 | 7 | 14 | 15 | 7 | 6 | 8 | 8 | |
NFTS | SES | 9 | 10 | 7 | 13 | 12 | 6 | 8 | 12 | 11 | |
NFTS | Holt | 8 | 9 | 9 | 14 | 13 | 9 | 6 | 10 | 4 | |
NFTS | ARIMA | 8 | 8 | 7 | 14 | 12 | 9 | 5 | 7 | 7 |
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Oancea, B.; Pospíšil, R.; Jula, M.N.; Imbrișcă, C.-I. Experiments with Fuzzy Methods for Forecasting Time Series as Alternatives to Classical Methods. Mathematics 2021, 9, 2517. https://doi.org/10.3390/math9192517
Oancea B, Pospíšil R, Jula MN, Imbrișcă C-I. Experiments with Fuzzy Methods for Forecasting Time Series as Alternatives to Classical Methods. Mathematics. 2021; 9(19):2517. https://doi.org/10.3390/math9192517
Chicago/Turabian StyleOancea, Bogdan, Richard Pospíšil, Marius Nicolae Jula, and Cosmin-Ionuț Imbrișcă. 2021. "Experiments with Fuzzy Methods for Forecasting Time Series as Alternatives to Classical Methods" Mathematics 9, no. 19: 2517. https://doi.org/10.3390/math9192517
APA StyleOancea, B., Pospíšil, R., Jula, M. N., & Imbrișcă, C.-I. (2021). Experiments with Fuzzy Methods for Forecasting Time Series as Alternatives to Classical Methods. Mathematics, 9(19), 2517. https://doi.org/10.3390/math9192517