# Fusing Nature with Computational Science for Optimal Signal Extraction

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## Abstract

**:**

## 1. Introduction

## 2. Basic SSA and SSA-CT

## 3. SSA-GA

- Run a basic SSA on training data and find the optimum r.
- Use the training data to build the trajectory matrix $\mathbf{X}={({x}_{ij})}_{i,j=1}^{L,K}=[{X}_{1},\dots ,{X}_{k}]$ where ${X}_{j}=({y}_{j},\dots ,{y}_{L+j-1}^{T})$.
- Apply the SVD for $\mathbf{X}$ and calculate eigenvalues ${\lambda}_{1}\ge \dots \ge {\lambda}_{L}$ and corresponding eigenvectors ${U}_{1},\cdots ,{U}_{L}$. Obtain ${V}_{i}={\mathbf{X}}^{T}{U}_{i}/\sqrt{{\lambda}_{i}}$ and ${\mathbf{X}}_{i}=\sqrt{{\lambda}_{i}}{U}_{i}{V}_{i}^{T}$.
- Define a chromosome ${C}_{i}$ as a vector of length L with binary values:$$\begin{array}{c}\hfill {C}_{i}=({c}_{i1},{c}_{i2},\dots ,{c}_{iL}),\end{array}$$
- Build a population containing M chromosomes, i.e., chromosomes ${C}_{1},\dots {C}_{M}$. Generate $K\%(K>70)$ of the chromosomes in the population randomly (from uniform distribution). This will produce chromosomes ${C}_{1}$ to ${C}_{k}$. Add ${C}_{k+1}=(0,0,\dots ,0)$ and ${C}_{k+2}=(1,1,\dots ,1)$ to the population (as extreme solutions). The rest of the population will be the same chromosomes as the basic SSA solution:$$\begin{array}{c}\hfill {c}_{ij}=\left(\right)open="\{"\; close>\begin{array}{cc}1\hfill & j\le r\hfill \\ 0\hfill & jr\hfill \end{array}i=k+3,\dots ,M,\end{array}$$
- Use a binary crossover function to produce ${M}^{\prime}$ offspring chromosomes. A simple crossover function produce offspring chromosomes as follows:
- (a)
- Pair chromosomes in the population randomly.
- (b)
- For a given pair of chromosomes ${C}_{i}$ and ${C}_{j}$ generate random number d from uniform distribution ($1\le d\le L$).
- (c)
- Produce offspring chromosomes for ${C}_{i}$ and ${C}_{j}$ with switching their first d genes:$$\begin{array}{ccc}& & \mathrm{First}\phantom{\rule{4.pt}{0ex}}\mathrm{offspring}=({c}_{i1},\dots ,{c}_{id},{c}_{j(d+1)},\dots ,{c}_{jL})\hfill \\ & & \mathrm{Sec}\mathrm{ond}\phantom{\rule{4.pt}{0ex}}\mathrm{offspring}=({c}_{j1},\dots ,{c}_{jd},{c}_{i(d+1)},\dots ,{c}_{iL})\hfill \end{array}$$

- Produce weight matrix ${\mathbf{W}}_{i}$ for each of $M+{M}^{\prime}$ chromosomes:$$\begin{array}{c}\hfill {\mathbf{W}}_{i}=\mathrm{diag}\left(\right)open="("\; close=")">{C}_{i},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,\dots ,M+{M}^{\prime}.\end{array}$$
- Reconstruct the signal for each weight matrix ${\mathbf{W}}_{i}$:$$\begin{array}{c}\hfill {\widehat{\mathbf{S}}}_{i}={\mathbf{U}}_{\mathbf{1}}\left(\right)open="("\; close=")">{\mathbf{W}}_{\mathbf{i}}{\Sigma}_{1}{\mathbf{V}}_{\mathbf{1}}^{\mathbf{T}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,\dots ,M+{M}^{\prime}.\end{array}$$
- For each chromosomes generate in-sample h step ahead forecasting and calculate the in-sample RMSE for all $M+{M}^{\prime}$ chromosomes. Select the M chromosomes with smallest RMSE as the new population.
- Repeat steps 6 to 9 until minimum RMSE in the population does not improve for several iterations.
- Begin with $L=2$ and repeat steps 1 to 10 for $2\le L\le \frac{N}{2}$, to find the L and grouping parameter which minimizes in-sample RMSE.

## 4. Empirical Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

Code | Name of Time Series |
---|---|

A001 | US Economic Statistics: Capacity Utilization. |

A002 | Births by months 1853–2012. |

A003 | Electricity: electricity net generation: total (all sectors). |

A004 | Energy prices: average retail prices of electricity. |

A005 | Coloured fox fur returns, Hopedale, Labrador, 1834–1925. |

A006 | Alcohol demand (log spirits consumption per head), UK, 1870–1938. |

A007 | Monthly Sutter county workforce, January 1946–December 1966 priesema (1979). |

A008 | Exchange rates—monthly data: Japanese yen. |

A009 | Exchange rates—monthly data: Pound sterling. |

A010 | Exchange rates—monthly data: Romanian leu. |

A011 | HICP (2005 = 100)—monthly data (annual rate of change): European Union (27 countries). |

A012 | HICP (2005 = 100)—monthly data (annual rate of change): UK. |

A013 | HICP (2005 = 100)—monthly data (annual rate of change): US. |

A014 | New Homes Sold in the United States. |

A015 | Goods, Value of Exports for United States. |

A016 | Goods, Value of Imports for United States. |

A017 | Market capitalisation—monthly data: UK. |

A018 | Market capitalisation—monthly data: US. |

A019 | Average monthly temperatures across the world (1701–2011): Bournemouth. |

A020 | Average monthly temperatures across the world (1701–2011): Eskdalemuir. |

A021 | Average monthly temperatures across the world (1701–2011): Lerwick. |

A022 | Average monthly temperatures across the world (1701–2011): Valley. |

A023 | Average monthly temperatures across the world (1701–2011): Death Valley. |

A024 | US Economic Statistics: Personal Savings Rate. |

A025 | Economic Policy Uncertainty Index for United States (Monthly Data). |

A026 | Coal Production, Total for Germany. |

A027 | Coke, Beehive Production (by Statistical Area). |

A028 | Monthly champagne sales (in 1000’s) (p. 273: Montgomery: Fore. and T.S.). |

A029 | Domestic Auto Production. |

A030 | Index of Cotton Textile Production for France. |

A031 | Index of Production of Chemical Products (by Statistical Area). |

A032 | Index of Production of Leather Products (by Statistical Area). |

A033 | Index of Production of Metal Products (by Statistical Area). |

A034 | Index of Production of Mineral Fuels (by Statistical Area). |

A035 | Industrial Production Index. |

A036 | Knit Underwear Production (by Statistical Area). |

A037 | Lubricants Production for United States. |

A038 | Silver Production for United States. |

A039 | Slab Zinc Production (by Statistical Area). |

A040 | Annual domestic sales and advertising of Lydia E, Pinkham Medicine, 1907 to 1960. |

A041 | Chemical concentration readings. |

A042 | Monthly Boston armed robberies January 1966–October 1975 Deutsch and Alt (1977). |

A043 | Monthly Minneapolis public drunkenness intakes January’66–July’78. |

A044 | Motor vehicles engines and parts/CPI, Canada, 1976–1991. |

A045 | Methane input into gas furnace: cu. ft/min. Sampling interval 9 s. |

A046 | Monthly civilian population of Australia: thousand persons. February 1978–April 1991. |

A047 | Daily total female births in California, 1959. |

A048 | Annual immigration into the United States: thousands. 1820–1962. |

A049 | Monthly New York City births: unknown scale. January 1946–December 1959. |

A050 | Estimated quarterly resident population of Australia: thousand persons. |

A051 | Annual Swedish population rates (1000’s) 1750–1849 Thomas (1940). |

A052 | Industry sales for printing and writing paper (in Thousands of French francs). |

A053 | Coloured fox fur production, Hebron, Labrador, 1834–1925. |

A054 | Coloured fox fur production, Nain, Labrador, 1834–1925. |

A055 | Coloured fox fur production, oak, Labrador, 1834–1925. |

A056 | Monthly average daily calls to directory assistance January’62–December’76. |

A057 | Monthly Av. residential electricity usage Iowa city 1971–1979. |

A058 | Montly av. residential gas usage Iowa (cubic feet)*100 ’71–’79. |

A059 | Monthly precipitation (in mm), January 1983–April 1994. London, United Kingdom. |

A060 | Monthly water usage (mL/day), London Ontario, 1966–1988. |

A061 | Quarterly production of Gas in Australia: million megajoules. Includes natural gas from July 1989. March 1956–September 1994. |

A062 | Residential water consumption, January 1983–April 1994. London, United Kingdom. |

A063 | The total generation of electricity by the U.S. electric industry (monthly data for the period January 1985–October 1996). |

A064 | Total number of water consumers, January 1983–April 1994. London, United Kingdom. |

A065 | Monthly milk production: pounds per cow. January 62–December 75. |

A066 | Monthly milk production: pounds per cow. January 62–December 75, adjusted for month length. |

A067 | Monthly total number of pigs slaughtered in Victoria. January 1980–August 1995. |

A068 | Monthly demand repair parts large/heavy equip. Iowa 1972–1979. |

A069 | Number of deaths and serious injuries in UK road accidents each month. January 1969–December 1984. |

A070 | Passenger miles (Mil) flown domestic U.K. July’62–May’72. |

A071 | Monthly hotel occupied room av. ’63–’76 B.L.Bowerman et al. |

A072 | Weekday bus ridership, Iowa city, Iowa (monthly averages). |

A073 | Portland Oregon average monthly bus ridership (/100). |

A074 | U.S. airlines: monthly aircraft miles flown (Millions) 1963–1970. |

A075 | International airline passengers: monthly totals in thousands. January 49–December 60. |

A076 | Sales: souvenir shop at a beach resort town in Queensland, Australia. January 1987–December 1993. |

A077 | Der Stern: Weekly sales of wholesalers A, ’71–’72. |

A078 | Der Stern: Weekly sales of wholesalers B, ’71–’72’ |

A079 | Der Stern: Weekly sales of wholesalers ’71–’72. |

A080 | Monthly sales of U.S. houses (thousands) 1965–1975. |

A081 | CFE specialty writing papers monthly sales. |

A082 | Monthly sales of new one-family houses sold in USA since 1973. |

A083 | Wisconsin employment time series, food and kindred products, January 1961–October 1975. |

A084 | Monthly gasoline demand Ontario gallon millions 1960–1975. |

A085 | Wisconsin employment time series, fabricated metals, January 1961–October 1975. |

A086 | Monthly empolyees wholes./retail Wisconsin ’61–’75 R.B.Miller. |

A087 | US monthly sales of chemical related products. January 1971–December 1991. |

A088 | US monthly sales of coal related products. January 1971–December 1991. |

A089 | US monthly sales of petrol related products. January 1971–December 1991. |

A090 | US monthly sales of vehicle related products. January 1971–December 1991. |

A091 | Civilian labour force in Australia each month: thousands of persons. February 1978–August 1995. |

A092 | Numbers on Unemployment Benefits in Australia: monthly January 1956–July 1992. |

A093 | Monthly Canadian total unemployment figures (thousands) 1956–1975. |

A094 | Monthly number of unemployed persons in Australia: thousands. February 1978–April 1991. |

A095 | Monthly U.S. female (20 years and over) unemployment figures 1948–1981. |

A096 | Monthly U.S. female (16–19 years) unemployment figures (thousands) 1948–1981. |

A097 | Monthly unemployment figures in West Germany 1948–1980. |

A098 | Monthly U.S. male (20 years and over) unemployment figures 1948–1981. |

A099 | Wisconsin employment time series, transportation equipment, January 1961–October 1975. |

A100 | Monthly U.S. male (16–19 years) unemployment figures (thousands) 1948–1981. |

Code | F | N | Mean | Med. | SD | CV | Skew. | SW(p) | ADF | Code | F | N | Mean | Med. | SD | CV | Skew. | SW(p) | ADF |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A001 | M | 539 | 80 | 80 | 5 | 6 | −0.55 | <0.01 | −0.60 ${}^{\u2020}$ | A002 | M | 1920 | 271 | 249 | 88 | 33 | 0.16 | <0.01 | −1.82 ${}^{\u2020}$ |

A003 | M | 484 | 2.59 × 10 ${}^{5}$ | 2.61 × 10 ${}^{5}$ | 6.88 × 10 ${}^{5}$ | 27 | 0.15 | <0.01 | −0.90 ${}^{\u2020}$ | A004 | M | 310 | 7 | 7 | 2 | 28 | −0.24 | <0.01 | 0.56 ${}^{\u2020}$ |

A005 | D | 92 | 47.63 | 31.00 | 47.33 | 99.36 | 2.27 | <0.01 | −3.16 | A006 | Q | 207 | 1.95 | 1.98 | 0.25 | 12.78 | −0.58 | <0.01 | 0.46 ${}^{\u2020}$ |

A007 | M | 252 | 2978 | 2741 | 1111 | 37.32 | 0.79 | <0.01 | −0.80 ${}^{\u2020}$ | A008 | M | 160 | 128 | 128 | 19 | 15 | 0.34 | <0.01 | −0.59 ${}^{\u2020}$ |

A009 | M | 160 | 0.72 | 0.69 | 0.10 | 13 | 0.66 | <0.01 | 0.53 ${}^{\u2020}$ | A010 | M | 160 | 3.41 | 3.61 | 0.83 | 24 | −0.92 | <0.01 | 1.58 ${}^{\u2020}$ |

A011 | M | 201 | 4.7 | 2.6 | 5.0 | 106 | 2.24 | <0.01 | −2.66 | A012 | M | 199 | 2.1 | 1.9 | 1.0 | 49 | 0.92 | <0.01 | −0.79 ${}^{\u2020}$ |

A013 | M | 176 | 2.5 | 2.4 | 1.6 | 66 | −0.52 | <0.01 | −2.27 ${}^{\u2020}$ | A014 | M | 606 | 55 | 53 | 20 | 35 | 0.79 | <0.01 | −1.41 ${}^{\u2020}$ |

A015 | M | 672 | 3.39 | 1.89 | 3.48 | 103 | 1.09 | <0.01 | 2.46 ${}^{\u2020}$ | A016 | M | 672 | 5.18 | 2.89 | 5.78 | 111 | 1.13 | <0.01 | 1.91 ${}^{\u2020}$ |

A017 | M | 249 | 130 | 130 | 24 | 19 | 0.35 | <0.01 | 0.24 ${}^{\u2020}$ | A018 | M | 249 | 112 | 114 | 25 | 22 | −0.01 | 0.01 * | 0.06 ${}^{\u2020}$ |

A019 | M | 605 | 10.1 | 9.6 | 4.5 | 44 | 0.05 | <0.01 | −4.77 | A020 | M | 605 | 7.3 | 6.9 | 4.3 | 59 | 0.04 | <0.01 | −6.07 |

A021 | M | 605 | 7.2 | 6.8 | 3.3 | 46 | 0.13 | <0.01 | −4.93 | A022 | M | 605 | 10.3 | 9.9 | 3.8 | 37 | 0.04 | <0.01 | −4.19 |

A023 | M | 605 | 24 | 24 | 10 | 40 | −0.02 | <0.01 | −7.15 | A024 | M | 636 | 6.9 | 7.4 | 2.6 | 38 | −0.29 | <0.01 | −1.18 ${}^{\u2020}$ |

A025 | M | 343 | 108 | 100 | 33 | 30 | 0.99 | <0.01 | −1.23 ${}^{\u2020}$ | A026 | M | 277 | 11.7 | 11.9 | 2.3 | 20 | −0.16 | 0.06 * | −0.40 ${}^{\u2020}$ |

A027 | M | 171 | 0.21 | 0.13 | 0.19 | 88 | 1.26 | <0.01 | −1.81 ${}^{\u2020}$ | A028 | M | 96 | 4801 | 4084 | 2640 | 54.99 | 1.55 | <0.01 | −1.66 ${}^{\u2020}$ |

A029 | M | 248 | 391 | 385 | 116 | 30 | −0.03 | 0.08 * | −1.22 ${}^{\u2020}$ | A030 | M | 139 | 89 | 92 | 12 | 13 | −0.82 | <0.01 | −0.28 ${}^{\u2020}$ |

A031 | M | 121 | 134 | 138 | 27 | 20 | 0.05 | <0.01 | 1.51 ${}^{\u2020}$ | A032 | M | 153 | 113 | 114 | 10 | 9 | −0.29 | 0.45 * | −0.52 ${}^{\u2020}$ |

A033 | M | 115 | 117 | 118 | 17 | 15 | −0.29 | 0.03 * | −0.46 ${}^{\u2020}$ | A034 | M | 115 | 110 | 111 | 11 | 10 | −0.53 | 0.02 * | 0.30 ${}^{\u2020}$ |

A035 | M | 1137 | 40 | 34 | 31 | 78 | 0.56 | <0.01 | 5.14 ${}^{\u2020}$ | A036 | M | 165 | 1.08 | 1.10 | 0.20 | 18.37 | −1.15 | <0.01 | −0.59 ${}^{\u2020}$ |

A037 | M | 479 | 3.04 | 2.83 | 1.02 | 33.60 | 0.46 | <0.01 | 0.61 ${}^{\u2020}$ | A038 | M | 283 | 9.39 | 10.02 | 2.27 | 24.15 | −0.80 | <0.01 | −1.01 ${}^{\u2020}$ |

A039 | M | 452 | 54 | 52 | 19 | 36 | −0.15 | <0.01 | 0.08 ${}^{\u2020}$ | A040 | Q | 108 | 1382 | 1206 | 684 | 49.55 | 0.83 | <0.01 | −0.80 ${}^{\u2020}$ |

A041 | H | 197 | 17.06 | 17.00 | 0.39 | 2.34 | 0.15 | 0.21 * | 0.09 ${}^{\u2020}$ | A042 | M | 118 | 196.3 | 166.0 | 128.0 | 65.2 | 0.45 | <0.01 | 0.41 ${}^{\u2020}$ |

A043 | M | 151 | 391.1 | 267.0 | 237.49 | 60.72 | 0.43 | <0.01 | −1.17 ${}^{\u2020}$ | A044 | M | 188 | 1344 | 1425 | 479.1 | 35.6 | −0.41 | <0.01 | −1.28 ${}^{\u2020}$ |

A045 | H | 296 | −0.05 | 0.00 | 1.07 | −1887 | −0.05 | 0.55 * | −7.66 | A046 | M | 159 | 11,890 | 11,830 | 882.93 | 7.42 | 0.12 | <0.01 | 5.71 |

A047 | D | 365 | 41.98 | 42.00 | 7.34 | 17.50 | 0.44 | <0.01 | −1.07 ${}^{\u2020}$ | A048 | A | 143 | 2.5 × 10${}^{5}$ | 2.2 × 10${}^{5}$ | 2.1 × 10${}^{5}$ | 83.19 | 1.06 | <0.01 | −2.63 |

A049 | M | 168 | 25.05 | 24.95 | 2.31 | 9.25 | −0.02 | 0.02 * | 0.07 ${}^{\u2020}$ | A050 | Q | 89 | 15,274 | 15,184 | 1358 | 8.89 | 0.19 | <0.01 | 9.72 ${}^{\u2020}$ |

A051 | A | 100 | 6.69 | 7.50 | 5.88 | 87.87 | −2.45 | <0.01 | −3.06 | A052 | M | 120 | 713 | 733 | 174 | 24.39 | −1.09 | <0.01 | −0.78 ${}^{\u2020}$ |

A053 | A | 91 | 81.58 | 46.00 | 102.07 | 125.11 | 2.80 | <0.01 | −3.44 | A054 | A | 91 | 101.80 | 77.00 | 92.14 | 90.51 | 1.43 | <0.01 | −3.38 |

A055 | A | 91 | 59.45 | 39.00 | 60.42 | 101.63 | 1.56 | <0.01 | −3.99 | A056 | M | 180 | 492.50 | 521.50 | 189.54 | 38.48 | −0.17 | <0.01 | −0.65 ${}^{\u2020}$ |

A057 | M | 106 | 489.73 | 465.00 | 93.34 | 19.06 | 0.92 | <0.01 | −1.21 ${}^{\u2020}$ | A058 | M | 106 | 124.71 | 94.50 | 84.15 | 67.48 | 0.52 | <0.01 | −3.88 |

A059 | M | 136 | 85.66 | 80.25 | 37.54 | 43.83 | 0.91 | <0.01 | −1.88 ${}^{\u2020}$ | A060 | M | 276 | 118.61 | 115.63 | 26.39 | 22.24 | 0.86 | <0.01 | −0.47 ${}^{\u2020}$ |

A061 | Q | 155 | 61,728 | 47,976 | 53,907 | 87.33 | 0.44 | <0.01 | 0.06 ${}^{\u2020}$ | A062 | M | 136 | 5.72 × 10${}^{7}$ | 5.53 × 10${}^{7}$ | 1.2 × 10${}^{7}$ | 21.51 | 1.13 | <0.01 | −0.84 ${}^{\u2020}$ |

A063 | M | 142 | 231.09 | 226.73 | 24.37 | 10.55 | 0.52 | 0.01 | −0.39 ${}^{\u2020}$ | A064 | M | 136 | 31,388 | 31,251 | 3232 | 10.30 | 0.25 | 0.22 * | −0.16 ${}^{\u2020}$ |

A065 | M | 156 | 754.71 | 761.00 | 102.20 | 13.54 | 0.01 | 0.04 * | 0.04 ${}^{\u2020}$ | A066 | M | 156 | 746.49 | 749.15 | 98.59 | 13.21 | 0.08 | 0.04 * | −0.38 ${}^{\u2020}$ |

A067 | M | 188 | 90,640 | 91,661 | 13,926 | 15.36 | −0.38 | 0.01 * | −0.38 ${}^{\u2020}$ | A068 | M | 94 | 1540 | 1532 | 474.35 | 30.79 | 0.38 | 0.05 * | 0.54 ${}^{\u2020}$ |

A069 | M | 192 | 1670 | 1631 | 289.61 | 17.34 | 0.53 | <0.01 | −0.74 ${}^{\u2020}$ | A070 | M | 119 | 91.09 | 86.20 | 32.80 | 36.01 | 0.34 | <0.01 | −1.93 ${}^{\u2020}$ |

A071 | M | 168 | 722.30 | 709.50 | 142.66 | 19.75 | 0.72 | <0.01 | −0.52 ${}^{\u2020}$ | A072 | W | 136 | 5913 | 5500 | 1784 | 30.17 | 0.67 | <0.01 | −0.68 ${}^{\u2020}$ |

A073 | M | 114 | 1120 | 1158 | 270.89 | 24.17 | −0.37 | <0.01 | 0.76 ${}^{\u2020}$ | A074 | M | 96 | 10,385 | 10,401 | 2202 | 21.21 | 0.33 | 0.18 * | −0.13 ${}^{\u2020}$ |

A075 | M | 144 | 280.30 | 265.50 | 119.97 | 42.80 | 0.57 | <0.01 | −0.35 ${}^{\u2020}$ | A076 | M | 84 | 14,315 | 8771 | 15,748 | 110 | 3.37 | <0.01 | −0.29 ${}^{\u2020}$ |

A077 | W | 104 | 11,909 | 11,640 | 1231 | 10.34 | 0.60 | <0.01 | −0.16 ${}^{\u2020}$ | A078 | W | 104 | 74,636 | 73,600 | 4737 | 6.35 | 0.64 | <0.01 | −0.59 ${}^{\u2020}$ |

A079 | W | 104 | 1020 | 1012 | 71.78 | 7.03 | 0.60 | 0.01 * | −0.41 ${}^{\u2020}$ | A080 | M | 132 | 45.36 | 44.00 | 10.38 | 22.88 | 0.17 | 0.15 * | −0.81 ${}^{\u2020}$ |

A081 | M | 147 | 1745 | 1730 | 479.52 | 27.47 | −0.39 | <0.01 | −1.15 ${}^{\u2020}$ | A082 | M | 275 | 52.29 | 53.00 | 11.94 | 22.83 | 0.18 | 0.13 * | −1.30 ${}^{\u2020}$ |

A083 | M | 178 | 58.79 | 55.80 | 6.68 | 11.36 | 0.93 | <0.01 | −0.92 ${}^{\u2020}$ | A084 | M | 192 | 1.62 × 10${}^{5}$ | 1.57 × 10${}^{5}$ | 41,661 | 25.71 | 0.32 | <0.01 | 0.25 ${}^{\u2020}$ |

A085 | M | 178 | 40.97 | 41.50 | 5.11 | 12.47 | −0.07 | <0.01 | 1.45 ${}^{\u2020}$ | A086 | M | 178 | 307.56 | 308.35 | 46.76 | 15.20 | 0.17 | <0.01 | 1.51 ${}^{\u2020}$ |

A087 | M | 252 | 13.70 | 14.08 | 6.13 | 44.73 | 0.16 | <0.01 | 1.13 ${}^{\u2020}$ | A088 | M | 252 | 65.67 | 68.20 | 14.25 | 21.70 | −0.53 | <0.01 | −0.53 ${}^{\u2020}$ |

A089 | M | 252 | 10.76 | 10.92 | 5.11 | 47.50 | −0.19 | <0.01 | −0.05 ${}^{\u2020}$ | A090 | M | 252 | 11.74 | 11.05 | 5.11 | 43.54 | 0.38 | <0.01 | −0.88 ${}^{\u2020}$ |

A091 | M | 211 | 7661 | 7621 | 819 | 10.70 | 0.03 | <0.01 | 3.27 ${}^{\u2020}$ | A092 | M | 439 | 2.21 × 10${}^{5}$ | 5.67 × 10${}^{4}$ | 2.35 × 10${}^{5}$ | 106.32 | 0.77 | <0.01 | 1.61 ${}^{\u2020}$ |

A093 | M | 240 | 413.28 | 396.50 | 152.84 | 36.98 | 0.36 | <0.01 | −1.60 ${}^{\u2020}$ | A094 | M | 211 | 6787 | 6528 | 604.62 | 8.91 | 0.56 | <0.01 | 2.69 ${}^{\u2020}$ |

A095 | M | 408 | 1373 | 1132 | 686.05 | 49.96 | 0.91 | <0.01 | 0.60 ${}^{\u2020}$ | A096 | M | 408 | 422.38 | 342.00 | 252.86 | 59.87 | 0.65 | <0.01 | −1.95 ${}^{\u2020}$ |

A097 | M | 396 | 7.14 × 10${}^{5}$ | 5.57 × 10${}^{5}$ | 5.64 × 10${}^{5}$ | 78.97 | 0.79 | <0.01 | −2.51 ${}^{\u2020}$ | A098 | M | 408 | 1937 | 1825 | 794 | 41.04 | 0.64 | <0.01 | −1.15 ${}^{\u2020}$ |

A099 | M | 178 | 40.60 | 40.50 | 4.95 | 12.19 | −0.65 | <0.01 | −0.10 ${}^{\u2020}$ | A100 | M | 408 | 520.28 | 425.50 | 261.22 | 50.21 | 0.64 | <0.01 | −1.65 ${}^{\u2020}$ |

^{†}indicates a nonstationary time series based on the Augmented Dickey-Fuller test at p = 0.01. A indicates annual, M indicates monthly, Q indicates quarterly, W indicates weekly, D indicates daily and H indicates hourly. N indicates series length.

Forecasting Horizon | ||||||||
---|---|---|---|---|---|---|---|---|

Series’ | h = 1 | h = 3 | h = 6 | h = 12 | ||||

Code | RRMSE | KSPA p-Value | RRMSE | KSPA p-Value | RRMSE | KSPA p-Value | RRMSE | KSPA p-Value |

A001 | 0.567 | 0.001 | 0.425 | 0.000 | 0.396 | 0.000 | 0.374 | 0.000 |

A002 | 1.297 | 0.001 | 1.347 | 0.000 | 1.358 | 0.000 | 1.359 | 0.000 |

A003 | 1.263 | 0.141 | 1.090 | 0.454 | 1.034 | 0.385 | 1.032 | 0.408 |

A004 | 1.632 | 0.017 | 0.543 | 0.012 | 0.547 | 0.105 | 0.572 | 0.391 |

A005 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A006 | 1.088 | 0.797 | 1.075 | 0.928 | 1.113 | 0.694 | 1.067 | 0.604 |

A007 | 2.015 | 0.092 | 2.452 | 0.026 | 3.339 | 0.162 | 667.713 | 0.042 |

A008 | 1.716 | 0.306 | 1.885 | 0.948 | 2.483 | 0.161 | 6.844 | 0.445 |

A009 | 0.976 | 1.000 | 0.969 | 0.948 | 0.966 | 0.997 | 0.973 | 1.000 |

A010 | 1.245 | 0.306 | 0.952 | 0.480 | 0.968 | 0.522 | 0.989 | 0.315 |

A011 | 0.669 | 0.231 | 0.371 | 0.068 | 0.427 | 0.050 | 0.616 | 0.043 |

A012 | 1.021 | 0.981 | 1.014 | 0.999 | 1.008 | 1.000 | 1.007 | 0.993 |

A013 | 1.027 | 1.000 | 1.010 | 1.000 | 1.008 | 1.000 | 1.010 | 0.999 |

A014 | 1.298 | 0.125 | 1.069 | 0.371 | 1.161 | 0.169 | 1.243 | 0.182 |

A015 | 2.259 | 0.000 | 1.663 | 0.000 | 1.121 | 0.114 | 0.901 | 0.000 |

A016 | 1.214 | 0.888 | 1.151 | 1.000 | 1.126 | 0.896 | 1.161 | 0.774 |

A017 | 0.614 | 0.132 | 0.587 | 0.063 | 0.672 | 0.414 | 0.700 | 0.717 |

A018 | 1.172 | 0.631 | 0.878 | 0.512 | 1.032 | 0.804 | 1.032 | 0.717 |

A019 | 1.021 | 0.518 | 1.012 | 0.254 | 1.030 | 0.534 | 1.025 | 0.870 |

A020 | 1.049 | 0.957 | 1.064 | 0.693 | 1.082 | 0.456 | 1.100 | 0.720 |

A021 | 1.117 | 0.984 | 1.154 | 0.524 | 1.148 | 0.383 | 1.141 | 0.720 |

A022 | 1.125 | 0.439 | 1.080 | 0.852 | 1.067 | 0.961 | 1.060 | 0.870 |

A023 | 1.135 | 0.518 | 1.150 | 0.374 | 1.158 | 0.319 | 1.128 | 0.720 |

A024 | 1.520 | 0.001 | 1.544 | 0.001 | 1.576 | 0.000 | 1.607 | 0.000 |

A025 | 1.796 | 0.003 | 1.729 | 0.002 | 1.661 | 0.006 | 1.796 | 0.000 |

A026 | 2.339 | 0.015 | 1.571 | 0.935 | 1.121 | 0.616 | 1.071 | 0.658 |

A027 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A028 | 1.081 | 0.987 | 1.054 | 0.422 | 1.062 | 0.501 | 1.072 | 0.160 |

A029 | 1.059 | 0.777 | 1.059 | 0.791 | 1.059 | 0.915 | 1.051 | 0.844 |

A030 | 1.627 | 0.707 | 6.047 | 0.545 | 49.035 | 0.420 | 3160.646 | 0.008 |

A031 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A032 | 1.172 | 0.919 | 1.103 | 0.791 | 1.208 | 0.653 | 1.453 | 0.562 |

A033 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A034 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A035 | 3.019 | 0.000 | 1.910 | 0.007 | 1.503 | 0.075 | 1.228 | 0.186 |

A036 | 1.079 | 0.994 | 1.083 | 0.996 | 1.062 | 0.964 | 1.061 | 0.979 |

A037 | 1.563 | 0.043 | 1.709 | 0.062 | 1.829 | 0.151 | 1.809 | 0.003 |

A038 | 0.820 | 0.724 | 0.757 | 0.940 | 0.936 | 0.631 | 0.953 | 0.892 |

A039 | 2.145 | 0.000 | 1.559 | 0.012 | 1.372 | 0.053 | 1.343 | 0.017 |

A040 | 1.104 | 0.996 | 1.073 | 0.997 | 1.038 | 0.999 | 1.035 | 1.000 |

A041 | 1.349 | 0.779 | 1.584 | 0.357 | 1.851 | 0.276 | 3.610 | 0.026 |

A042 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A043 | 1.068 | 0.919 | 1.036 | 0.932 | 1.011 | 0.995 | 0.969 | 0.877 |

A044 | 1.161 | 0.750 | 1.244 | 0.223 | 1.863 | 0.357 | 3.430 | 0.001 |

A045 | 1.745 | 0.000 | 1.476 | 0.000 | 1.474 | 0.000 | 1.521 | 0.016 |

A046 | 0.249 | 0.000 | 0.358 | 0.000 | 0.498 | 0.001 | 0.588 | 0.000 |

A047 | 0.923 | 0.438 | 0.877 | 0.663 | 0.823 | 0.224 | 0.786 | 0.403 |

A048 | 0.657 | 0.903 | 0.577 | 0.919 | 1.317 | 0.938 | 4.049 | 0.861 |

A049 | 1.042 | 0.653 | 1.099 | 0.680 | 1.109 | 0.967 | 1.273 | 0.622 |

A050 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A051 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A052 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A053 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A054 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A055 | 1.064 | 0.983 | 1.036 | 0.990 | 1.017 | 0.710 | 1.008 | 0.615 |

A056 | 1.346 | 0.536 | 1.166 | 0.728 | 1.116 | 0.897 | 1.089 | 0.928 |

A057 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A058 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A059 | 1.059 | 0.874 | 1.044 | 1.000 | 1.056 | 0.990 | 1.044 | 0.996 |

A060 | 1.460 | 0.124 | 1.439 | 0.133 | 1.353 | 0.374 | 1.317 | 0.174 |

A061 | 5.321 | 0.000 | 3.518 | 0.012 | 2.806 | 0.004 | 1.565 | 0.000 |

A062 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A063 | 1.061 | 0.528 | 0.816 | 0.562 | 0.765 | 0.610 | 0.892 | 0.100 |

A064 | 0.815 | 0.690 | 0.788 | 1.000 | 0.850 | 0.579 | 0.423 | 0.834 |

A065 | 0.931 | 0.926 | 0.888 | 0.994 | 0.887 | 0.996 | 0.892 | 0.739 |

A066 | 2.509 | 0.008 | 2.219 | 0.039 | 1.736 | 0.229 | 1.189 | 0.739 |

A067 | 1.085 | 0.750 | 0.876 | 0.610 | 0.733 | 0.249 | 0.490 | 0.036 |

A068 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A069 | 1.063 | 0.975 | 1.038 | 0.999 | 1.037 | 0.644 | 1.028 | 0.704 |

A070 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A071 | 0.874 | 0.653 | 0.858 | 0.680 | 0.845 | 0.396 | 0.883 | 0.999 |

A072 | 0.910 | 0.979 | 0.869 | 0.723 | 0.903 | 0.766 | 0.946 | 0.952 |

A073 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A074 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A075 | 1.389 | 0.135 | 1.523 | 0.389 | 1.463 | 0.791 | 1.352 | 0.693 |

A076 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A077 | 1.089 | 0.915 | 1.084 | 0.997 | 1.172 | 0.808 | 1.373 | 0.707 |

A078 | 1.105 | 0.996 | 1.084 | 0.997 | 1.095 | 0.958 | 1.083 | 0.707 |

A079 | 0.970 | 0.915 | 1.018 | 0.754 | 1.105 | 0.958 | 1.235 | 0.707 |

A080 | 1.081 | 0.999 | 1.055 | 0.979 | 1.032 | 1.000 | 1.032 | 1.000 |

A081 | 1.084 | 0.987 | 1.068 | 0.990 | 1.092 | 0.938 | 1.086 | 0.861 |

A082 | 0.977 | 0.998 | 1.040 | 0.844 | 1.059 | 0.744 | 1.049 | 0.991 |

A083 | 0.781 | 0.383 | 0.847 | 0.562 | 0.900 | 0.977 | 0.592 | 0.665 |

A084 | 0.634 | 0.004 | 0.544 | 0.033 | 0.396 | 0.041 | 0.054 | 0.002 |

A085 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A086 | 2.643 | 0.000 | 2.357 | 0.013 | 2.123 | 0.214 | 1.715 | 0.182 |

A087 | 2.030 | 0.023 | 1.758 | 0.147 | 1.552 | 0.162 | 1.293 | 0.268 |

A088 | 0.907 | 0.996 | 0.982 | 0.791 | 1.031 | 0.915 | 1.129 | 0.268 |

A089 | 1.868 | 0.198 | 1.255 | 0.791 | 1.006 | 0.546 | 0.924 | 0.194 |

A090 | 1.132 | 0.968 | 1.099 | 0.997 | 1.120 | 0.810 | 1.118 | 0.594 |

A091 | 0.507 | 0.513 | 0.132 | 0.131 | 0.009 | 0.098 | 0.000 | 0.084 |

A092 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A093 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

A094 | 3.369 | 0.000 | 2.508 | 0.019 | 2.094 | 0.098 | 1.827 | 0.020 |

A095 | 3.167 | 0.000 | 2.090 | 0.004 | 1.724 | 0.016 | 1.541 | 0.004 |

A096 | 1.692 | 0.020 | 1.696 | 0.086 | 1.798 | 0.024 | 1.867 | 0.006 |

A097 | 1.188 | 0.591 | 1.077 | 0.957 | 1.039 | 0.723 | 1.049 | 0.287 |

A098 | 0.587 | 0.059 | 0.649 | 0.015 | 0.765 | 0.352 | 0.902 | 0.563 |

A099 | 0.780 | 0.261 | 0.838 | 0.877 | 0.897 | 0.598 | 0.953 | 0.987 |

A100 | 0.947 | 0.612 | 0.736 | 0.086 | 0.729 | 0.278 | 0.720 | 0.078 |

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**Figure 2.**Histogram of RRMSEs for different forecasting horizons (To better illustrate the data, one extreme value is removed for h = 6 and two extreme values are removed for h = 12).

Factor | Levels | |||||
---|---|---|---|---|---|---|

Annual | Monthly | Quarterly | Weekly | Daily | Hourly | |

Sampling Frequency | 5 | 83 | 4 | 4 | 2 | 2 |

Positive Skew | Negative Skew | Symmetric | ||||

Skewness | 61 | 21 | 18 | |||

Normal | Non-normal | |||||

Normality | 18 | 82 | ||||

Stationary | Non-Stationary | |||||

Stationarity | 14 | 86 |

Forecasting Horizon | |||||
---|---|---|---|---|---|

h = 1 | h = 3 | h = 6 | h = 12 | ||

RRMSE’s Median | 1.0618 | 1.0362 | 1.0319 | 1.0302 | |

N. RRMSE < 1 ${}^{1}$ | 21 | 24 | 21 | 24 | |

N. RRMSE > 1 ${}^{2}$ | 57 | 54 | 57 | 54 | |

N. RRMSE < 1 (Significantly) ${}^{3}$ | 3 | 5 | 4 | 6 | |

N. RRMSE > 1 (Significantly) ${}^{3}$ | 17 | 13 | 7 | 14 | |

RRMSE ∼ Frequency ${}^{4}$ | 0.1975 | 0.1975 | 0.1975 | 0.1975 | |

Kruskal-Wallis | RRMSE ∼ Normality ${}^{4}$ | 0.9047 | 0.9047 | 0.9047 | 0.9047 |

p-value’s | RRMSE ∼ Stationarity ${}^{4}$ | 0.1625 | 0.1625 | 0.1625 | 0.1625 |

RRMSE ∼ Skewness ${}^{4}$ | 0.9618 | 0.9618 | 0.9618 | 0.9618 |

^{1}Number of RRMSEs less than 1;

^{2}Number of RRMSEs larger than 1;

^{3}Cases with KSPA’s p-value less than 0.05;

^{4}Kruskal-Wallis’ p-value for testing the effect of given factor on RRMSE.

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**MDPI and ACS Style**

Hassani, H.; Yeganegi, M.R.; Huang, X.
Fusing Nature with Computational Science for Optimal Signal Extraction. *Stats* **2021**, *4*, 71-85.
https://doi.org/10.3390/stats4010006

**AMA Style**

Hassani H, Yeganegi MR, Huang X.
Fusing Nature with Computational Science for Optimal Signal Extraction. *Stats*. 2021; 4(1):71-85.
https://doi.org/10.3390/stats4010006

**Chicago/Turabian Style**

Hassani, Hossein, Mohammad Reza Yeganegi, and Xu Huang.
2021. "Fusing Nature with Computational Science for Optimal Signal Extraction" *Stats* 4, no. 1: 71-85.
https://doi.org/10.3390/stats4010006