A Study on the Rapid Parameter Estimation and the Grey Prediction in Richards Model

2.2.1 Three-Stage Method

The parameters in Richards model can be estimated using the three-stage method[10]. The formula is derived as follows:

Take the logarithms on both sides of Equation (1), then derive them, the expression is 1ytdytdt=−1δeβ−rt(−r)1+eβ−rt.Setτt=1ydytdt, then

Equation (2) shows that the relative growth rate of Richards model curve τ_t, its reciprocal increases as time changes and follows the revised index rule.

The time sequence on the horizontal axis is divided into three equal length n. Set the boundary points on horizontal axis as t₁, t₂, t₃, the correspondences of Richard model curve on the vertical axis are y₁, y₂, y₃. The sum of 1τt in the three equal length n respectively is ∑11τt,∑21τt,∑31τt

Since M₁ (t₁, y₁), M₂ (t₂, y₂), M₃ (t₃, y₃) are valid in Equation (1), and t₂ − t₁ = t₃ − t₂ = n, thus, y1−δ−αδy2−δ−αδ=y2−δ−αδy3−δ−αδ=ern. After calculations, there are

Equations (3) to (6) is the estimation formula of the parameters in Richards model using three-stage method.

2.2.2 Optimization by Nonlinear Least Squares Method

With the experimental data (x_t, y_t) (t = 1, 2, ⋯, N), finding function f(x, θ) that makes min∑t=1N(f(xt,θ)−yt)2=∑t=1N(f(xt,φ)−yt)2θ is the parameter to be determined, and φ is the best parameter that is determined by the least squares[11, 12].

If f(x, θ) is a polynomial function, it is called a polynomial regression; if it is exponential, logarithmic, exponential, trigonometric or other functions, it is called nonlinear fitting.

Estimating the parameters of Richards model, the three-stage method is used to determine the initial parameters, then nonlinear least squares to optimize them. When the values meet the conditions of “coefficients of determination”, they are considered as the optimized values. The calculations for the “coefficient of determination”[13 – 14] is R2=1−∑t=1N(yt−yφ)2∑t=1N(yt−yp)2 where yp=1N∑t=1Nyt. The nearer R² is to 1, the better the curve fitting.

2.3.1 The “Kernel” of the Interval Grey Number

In grey system, the grey number[16] is referred as a quantity with only its possible range known but the exact value is unknown. Interval grey number is the most common form of grey information in the grey system.

Set ⊗ ∈ [a1, a2], a1 < a2, a1 and a2 are real number, they are respectively the lower and upper limit of ⊗, while the range of ⊗ is unknown. 1) If ⊗ is a continuous grey number, then ⊗=12(a1+a2) is the “Kernel” of the grey number; 2) If ⊗ is a discrete grey number, a_i∈ [a1, a2] (i = 1, 2, ⋯, n) represents all possible values of the grey number ⊗, therefore ⊗=1n(a1+a2) is the “Kernel” of the grey number ⊗[17–18].

2.3.2 The Generation of Numbers

Through processing the data in the sequence, a new sequence is generated; using this principle to explore the regularity among numbers is called the generation of numbers. There are many ways that numbers are generated: Accumulated generation, inverse accumulated generation, weighted accumulated generation, etc.[19].

When building the grey data for prediction, the accumulated (or inverse accumulated) generation can be considered as the data development tendency, from which the regularities inside the original data may emerge.

Accumulating generation operation (AGO)[20] is defined as follows:

The original time sequence with n samples is x⁰ = (x⁽⁰⁾ (1), (x⁽⁰⁾ (2), ⋯, (x⁽⁰⁾ (n)).

Construct monotonic increasing sequence x⁽¹⁾ by a one-time accumulated generating progression, expressed as x⁽¹⁾ = (x⁽¹⁾ (1),(x⁽¹⁾(2), ⋯,(x⁽¹⁾ (n)).Where x(1)(k)=∑i=1kx(0)(i),k = 1, 2, ⋯, n. x⁽¹⁾ is called one-time AGO.

If x⁽¹⁾ = x⁽¹⁾ (k) − x⁽¹⁾ (k − 1), k = 1, 2, ⋯, n, then x⁽¹⁾ is called a one-time inverse accumulated generating operation (IAGO).

3.1.1 Dividing Time Sequence into Three Equal-Length Stages and Determining the Value t₀

Figure 2 is the data analysis on “Growth of Rice Leaves”.

Figure 2

Data analysis on “Growth of Rice Leaves”

There are 21 original data of “Growth of Rice Leaves”: y(k), k = 1, 2, ⋯, 21. The time sequence on the horizontal axis is divided into three stages of equal length n (21/3 = 7, n = 7), the corresponding division points on the vertical axis value are y(1), y(7); y(8), y(14); y(15), y(21).

Δk = (k + 1) − k = 1, the derivative of y(k): [y(k + 1) − y(k)]/Δk = y(k + 1) − y(k), as shown by the dotted line in Figure 2. Observing, the dotted line “y(k + 1) − y(k)” indicating the three stages can be distinguished:

1. It starts with a stage where the “y(k+1) − y(k)” are relatively low, revealing that y(k) increases slowly, that is “the budding stage”.

2. In next stage, the “y(k + 1) − y(k)” increases and fluctuation, its average value increases remarkably comparing to the first stage. It means that y(k) increases rapidly, that is “the rapid growth stage”.

3. In last stage, the “y(k + 1) − y(k)” gradually decreases, revealing that y(k) increase slowly, that is “the saturation stage”.

Figure 2 is a typical Richards model curve with three stages. There is another typical Richards model curve only with, “fast growth” and “saturation” two stages. We can distinguish them as the value of “y(k + 1) − y(k)” changes.

According to the definition, t₀ is the value on the time axis when y_t = 0. However, sometimes errors can appear in the experiments when t₀ is calculated in three-stage method. Therefore, in this paper t₀ is determined by the empirical parameters.

If the data starts from the budding stage, and the number of data in each stage is less than 20 (the case when more than 20 data is involved is too complicated thus not discussed in this paper), t₀ can take a value among −14 to −20, then according to −e^β = e^−rt₀, calculate β. If the data starts from “rapid growth stage”, a “virtual budding stage” needs to be created, which the data begins at 0. The length of the “virtual budding stage” can be considered as half the sum of rapid growth stage and saturation stage, so that t₀ can be determined.

3.1.2 The Parameters are Estimated Rapidly by Arithmetic Sequence and Optimized by Nonlinear Least Squares

As shown in Figure 2 and Figure 1 A, the dividing points of the stages (original data) are connected, and the trend line of each stage is obtained: y(1)y(7), y(8)y(14), y(15)y(21).

The intersecting point of the trend line and the equidistant timeline can be referred as an approximate point of the original data. Obviously, the intersecting points of every stage can form an arithmetic sequence with common difference d, therefore there are the following equations:

Since τt=1ydytdt its discrete form is τk=1y(k)[y(k+1)−y(k)] therefore,

Combining Equation (7), (8), and (9), there is

in which d1=y(n)−y(1)n−1; Similarly,

in which d2=y(2n)−y(n+1)n−1;

in which d3=y(3n)−y(2n+1)n−1.

Put the above values ∑11τt,∑21τt,∑31τt into Equations (3), (4), (6) to calculate the values of r, δ, α; the value of β is calculated by −e^β = e^−rt₀.

Use the above parameters as the initial values to fit the original data by nonlinear least squares, when the “coefficients of determination” satisfies the predetermined conditions, optimized parameters of Richards model are obtained.

3.3.1 The Experimental Data of Rapid Parameter Estimation in Richards Model

As shown in Table 1, the comparison between the optimized value and the initial value in Richards model for the two practical examples: “Growth of Rice Leaves” and “Amoeba Cell Growth”. Table 1 and Table 2, the data errors are found within a reasonable range. It is also shown by the fitted curve A and B in Table 2, which is relatively accurate. Therefore, the algorithm of “rapid parameter estimation in Richards model” presented in this paper is feasible.

3.3.2 The Experimental Data of Grey Prediction in Richards Model

As shown in Table 3, the comparison between the optimized value and the initial value in the grey Richards model based on two practical examples: “Growth of Rice Leaves” and “Shopping Market Transaction Scale in China”. The data errors are within a reasonable range.

In Table 4A: Known original data are y(k), k = 1, 2, ⋯, 12. The fitted data are y(k), k = 13, 14, ⋯, 21. The data errors are within a reasonable range.

In Table 4B, the data need to be explained as follows.

Table 4B shows the data prediction in the grey Richards model and the fitted curve of “Shopping Market Transaction Scale in China” in the period of 2007Q4 to 2023Q1 (year and quarter, for example: 2007Q4 means “the fourth quarter of year 2007”), corresponding y(k), k = 1, 2, ⋯, 62.

Among them “y(1), y(2), ⋯, y(26)” are known data; fitted data are y(27), y(28), y(29), y(30) to verify the accuracy of the grey prediction in Richards model; the rest of the data are on the future development, for reference only. y(27), y(28), y(29), y(30) corresponding to 2014Q2data, 2014Q3data, 2014Q4data, 2015Q1data.

In the experiment, periodic fluctuations are observed during original data in Richards model, sometimes. For two sets of original data in the same stage on Richards model curve, their patterns of fluctuation are approximately the same. This rule can be used in the grey Richards model prediction.

As the “Double 11 Shopping Festival” since 2012 and the end of year shopping boom, y(k) rises to a remarkable increase in Q4; while most logistic companies do not work at the beginning of the year (the Spring Festival), there is a remarkable decrease y(k) in Q1. The periodic fluctuation patterns are marked by dotted rectangles W1 and W2 in Table 4 B, they are all in the “rapid growth stage” of the grey Richards model curve. So the fitted data in W3 need to be corrected according to the fluctuation pattern in W1 and W2, the process is shown by Equation (13), (14).

2013Q4data and 2013Q3data are known, 2014Q3data is obtained by curve fitting in Richards model, so 2014Q4data can be calculated by Equation (13).

As 2014Q1data and 2013Q4data are known, 2015Q1data can be calculated by Equation (14). So 2014Q4data and 2015Q1data replace the corresponding fitting data in Richards model curve, realizing the correction.

In Table 4B, the data errors are within a reasonable range, we can also find from the “fitted curve A, B” in Table 4, the fitted curve is relatively accurate. Therefore, the “grey prediction algorithm in Richards model” presented in this paper performs well.