# Wavelet Time-Scale Modeling of Brand Sales and Prices

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Wavelets Toolkit

- The j-level detail space ${W}_{j}=span({\psi}_{j,k},\phantom{\rule{0.166667em}{0ex}}k)$;
- The j-level approximation space ${V}_{j}=span({\phi}_{j,k},\phantom{\rule{0.166667em}{0ex}}k)$.

## 4. Development of the Mathematical Modeling System

- ${S}_{t}$ stands for the sales value at time t;
- ${P}_{t}$ stands for the price at time t;
- $C{P}_{t}$ stands for the competitor’s price at time t;
- $P{R}_{t}$ is the PROMO variable at time t;
- ${D}_{t}$ refers to percentage distribution of the main brand;
- ${\u03f5}_{t}$ is an error term.

- ${S}_{t,j}$ reflects the sales value at time t and the level or horizon j;
- ${P}_{t,j}$ stands for the price at level j and time t;
- $C{P}_{t,j}$ stands for the competitor’s price at level j and time t;
- $P{R}_{t,j}$ is the PROMO variable at horizon j and time t;
- ${D}_{t,j}$ stands for the percentage distribution at horizon j and time t;
- ${\u03f5}_{t,j}$ is an error term already relative to level j and time t.

## 5. Results and Discussion

**Figure 13.**The wavelet approximation ${A}_{6}$ for ${B}_{1}$ and $CP{B}_{1}$ prices and sales at level 6.

**Figure 14.**The wavelet approximations ${A}_{6}$ for ${B}_{2}$ and $CP{B}_{2}$ prices and sales at level 6.

**Figure 15.**The wavelet approximation ${A}_{6}$ for ${B}_{3}$ and $CP{B}_{3}$ prices and sales at level 6.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Nomination | Brand | Description (Sector) |
---|---|---|

${B}_{1}$ | Jarir bookstore | Books and electronics |

${B}_{2}$ | Almarai | Dairy and poultry |

${B}_{3}$ | STC | Telecommunications |

${B}_{4}$ | Al Abdullatif | Household durables |

${B}_{5}$ | EIC | Electrical industries company |

${B}_{6}$ | Al Aseel | Textiles, apparel and luxury goods |

Brand | Mean | Median | Min | Max | Std | Skewness | Kurtosis | Jarque–Bera $(\mathit{h},\mathit{p})$ |
---|---|---|---|---|---|---|---|---|

The brand prices | ||||||||

${B}_{1}$ | 161.42 | 159.2 | 105.44 | 225 | 26.48 | 0.21 | 2.37 | (1,${10}^{-3}$) |

${B}_{2}$ | 52.91 | 53.3 | 36.95 | 63.7 | 4.07 | −0.08 | 3.15 | (0,0.34) |

${B}_{3}$ | 100.25 | 99.75 | 67.10 | 139.2 | 16.35 | 0.20 | 2.52 | (1,10${}^{-3}$) |

${B}_{4}$ | 15.88 | 12.66 | 8.15 | 40.35 | 8.12 | 1.76 | 4.64 | (1,10${}^{-3}$) |

${B}_{5}$ | 20 | 20.34 | 14.46 | 29.4 | 3.55 | 0.39 | 2.35 | (1,10${}^{-3}$) |

${B}_{6}$ | 35.52 | 31.35 | 14.88 | 69.75 | 16.44 | 0.65 | 2.01 | (1,10${}^{-3}$) |

The brands sales | ||||||||

${B}_{1}$ | 153.63 | 109.31 | 7.56 | 4020 | 258.03 | 10.70 | 140.71 | (1,10${}^{-3}$) |

${B}_{2}$ | 564.28 | 418.03 | 34.70 | 12,140 | 689.47 | 9.25 | 129.83 | (1,10${}^{-3}$) |

${B}_{3}$ | 890.12 | 512.49 | 33.30 | 1.21310 | 3967.75 | 27.64 | 831.52 | (1,10${}^{-3}$) |

${B}_{4}$ | 809.01 | 223.43 | 8.24 | 22,200 | 1935.15 | 6.08 | 51.00 | (1,10${}^{-3}$) |

${B}_{5}$ | 2577.69 | 1720 | 111.39 | 37,490 | 3161.13 | 4.89 | 40.35 | (1,10${}^{-3}$) |

${B}_{6}$ | 243.89 | 64.04 | 0.01 | 5250 | 570.35 | 4.70 | 29.97 | (1,10${}^{-3}$) |

Brand | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|

The brand prices | ||||||

${B}_{1}$ | −0.0002 | 0.0007 | 0.0016 | 0.0147 | 0.0147 | 0.0912 |

${B}_{2}$ | 0.0000 | 0.0000 | −0.0001 | −0.0054 | −0.0054 | −0.0041 |

${B}_{3}$ | −0.0000 | −0.0004 | 0.0001 | 0.0042 | 0.0042 | 0.0656 |

${B}_{4}$ | −0.0000 | −0.0001 | −0.0008 | 0.0001 | 0.0001 | −0.0106 |

${B}_{5}$ | −0.0000 | 0.0002 | −0.0007 | −0.0045 | −0.0045 | −0.0061 |

${B}_{6}$ | 0.0000 | 0.0002 | −0.0008 | 0.0054 | 0.0054 | −0.0395 |

The brand sales | ||||||

${B}_{1}$ | −0.0073 | 0.0196 | −0.0206 | 0.0079 | 0.0079 | 0.0034 |

${B}_{2}$ | −0.0645 | −0.0049 | 0.0065 | 0.2211 | 0.2211 | 0.7129 |

${B}_{3}$ | 0.0012 | −0.4976 | −0.9052 | 36.9157 | 36.9157 | 20.2465 |

${B}_{4}$ | −0.2039 | 0.2151 | −0.0401 | 0.3270 | 0.3270 | 1.0747 |

${B}_{5}$ | −0.2954 | 0.2021 | −1.5023 | 4.1040 | 4.1040 | 21.1156 |

${B}_{6}$ | −0.0060 | 0.0060 | −0.0210 | 0.4643 | 0.4643 | 0.5981 |

Brands | Top Competitor(s) |
---|---|

${B}_{1}$—Jarir Bookstore | Alobeikan ($CP{B}_{1}$) |

${B}_{2}$—Almarai | Nadec ($CP{B}_{2}$) |

${B}_{3}$—STC | Zain ($CP{B}_{3}$) |

**Table 5.**Model (9) coefficient estimations for brands ${B}_{i}$, $1\le i\le 6$, at different levels.

Coefficient | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

The brand ${B}_{1}$ | ||||||||

${\beta}_{0}$ | $-95.21$ | $-167.57$ | $-0.01$ | $0.01$ | $-0.01$ | $-0.02$ | $-0.09$ | $-0.24$ |

${\beta}_{1}$ | $1.93$ | $2.45$ | $1.27$ | $12.75$ | $4.55$ | $2.85$ | $-7.13$ | $-4.82$ |

${\beta}_{2}$ | $-6.05$ | $-7.21$ | $-10.78$ | $-8.27$ | $15.78$ | $-24.60$ | $-3.54$ | $-12.48$ |

The brand ${B}_{2}$ | ||||||||

${\beta}_{0}$ | $-42.76$ | $-131.92$ | $-0.06$ | $0.02$ | $-0.002$ | $-0.42$ | $0.21$ | $1.10$ |

${\beta}_{1}$ | $10.31$ | $12.98$ | $227.21$ | $166.02$ | $-10.61$ | $45.20$ | $-6.22$ | $-34.84$ |

${\beta}_{2}$ | $2.17$ | $0.31$ | $53.16$ | $-72.86$ | $-14.65$ | $60.14$ | $-6.97$ | $23.62$ |

The brand ${B}_{3}$ | ||||||||

${\beta}_{0}$ | $139.14$ | $72.38$ | $-0.006$ | $-0.45$ | $-0.82$ | $-14.90$ | $35.88$ | $21.80$ |

${\beta}_{1}$ | $-5.91$ | $-7.47$ | $-590.90$ | $80.18$ | $-314.70$ | $41.34$ | $01.99$ | $16.75$ |

${\beta}_{2}$ | $122.77$ | $139.51$ | $2436.79$ | $-396.59$ | $351.10$ | $-394.74$ | $-980$ | $-167.01$ |

**Table 6.**Confidence intervals at $95\%$ for model (9) coefficient estimations for brands ${B}_{i}$, $1\le i\le 6$, at different levels.

Coeff | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

The brand ${B}_{1}$ | ||||||||

${\beta}_{0}$ | [−96.24; −95.18] | [−168.60; −165.53] | [−0.04; 0.02] | [−0.02; 0.04] | [−0.04; 0.02] | [−0.05; 0.02] | [−0.12; −0.08] | [−0.27; −0.21] |

${\beta}_{1}$ | [1.89; 1.96] | [2.41; 2.48] | [1.23; 1.30] | [12.71; 12.78] | [4.51; 4.58] | [2.81; 2.88] | [−7.16; −7.09] | [−4.85; −4.78] |

${\beta}_{2}$ | [−6.11; −6.00] | [−7.27; −7.16] | [−10.84; −10.73] | [−8.33; −8.22] | [15.71; 15.82] | [−24.66; −24.55] | [−3.60; −3.49] | [−12.54; −12.43] |

The brand ${B}_{2}$ | ||||||||

${\beta}_{0}$ | [−43.79; −42.77] | [−132.95; −130.88] | [−0.19; 0.02] | [−0.11; 0.05] | [−0.03; 0.02] | [−0.55; −0.30] | [0.10; 0.30] | [1.01; 1.23] |

${\beta}_{1}$ | [10.27; 10.34] | [12.94; 13.01] | [227.17; 227.24] | [165.98; 166.05] | [−10.64; −10.57] | [45.16; 45.23] | [−6.25; −6.18] | [−34.87; −34.80] |

${\beta}_{2}$ | [2.10; 2.22] | [0.24; 0.36] | [53.09; 53.21] | [−72.92; −72.80] | [−14.71; −14.59] | [60.07; 60.19] | [−7.03; −6.91] | [23.55; 23.67] |

The brand ${B}_{3}$ | ||||||||

${\beta}_{0}$ | [139.10; 139.17] | [72.34; 72.41] | [−0.03; 0.02] | [−0.48; −0.41] | [−0.85; −0.78] | [−14.93; −14.86] | [35.84; 35.91] | [21.76; 21.83] |

${\beta}_{1}$ | [−5.94; −5.87] | [−7.50; −7.43] | [−590.93; −590.86] | [80.14; 80.21] | [−314.73; −314.66] | [41.30; 41.37] | [1.95; 2.02] | [16.71; 16.78] |

${\beta}_{2}$ | [122.70; 122.81] | [139.44; 139.55] | [2436.72; 2436.83] | [−396.65; −396.54] | [351.03; 351.14] | [−394.80; −394.69] | [−980.06; −979.95] | [−167.07; −166.96] |

Brand | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

${B}_{1}$ | 0.26 | 0.58 | 0.17 | 0.12 | 0.78 | 0.55 | 0.56 | 0.23 |

${B}_{2}$ | 0.68 | 0.19 | 0.40 | 0.33 | 0.23 | 0.21 | 0.18 | 0.10 |

${B}_{3}$ | 0.39 | 0.63 | 0.30 | 0.14 | 0.12 | 0.99 | 0.65 | 0.23 |

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**MDPI and ACS Style**

Alanazi, T.M.; Ben Mabrouk, A.
Wavelet Time-Scale Modeling of Brand Sales and Prices. *Appl. Sci.* **2022**, *12*, 6485.
https://doi.org/10.3390/app12136485

**AMA Style**

Alanazi TM, Ben Mabrouk A.
Wavelet Time-Scale Modeling of Brand Sales and Prices. *Applied Sciences*. 2022; 12(13):6485.
https://doi.org/10.3390/app12136485

**Chicago/Turabian Style**

Alanazi, Tawfeeq M., and Anouar Ben Mabrouk.
2022. "Wavelet Time-Scale Modeling of Brand Sales and Prices" *Applied Sciences* 12, no. 13: 6485.
https://doi.org/10.3390/app12136485