Restricted Inference in Circular-Linear and Linear-Circular Regression

In this paper, we investigate restricted inference on two types of circular regression, called circular-linear and linear-circular. Our aim in this paper is to propose an alternative method which is necessary to apply where one observes a weak association between circular dependent and linear predictor variables, or between linear dependent and circular predictor variables, having clear knowledge about the sign of slope. We illustrate that restricted inference is particularly useful for those circular regressions, which is due to weak association. Comparison between our proposed restricted inference and the unrestricted inference are given by using two examples, one from ecological study and the other from environmental study.


Introduction
Circular variables are those that take any periodic measurements. Two typical examples are angle, which is periodic in 360 degrees, and the hourly time, which is periodic in 24 hours. Circular variables appear in many areas of research. Various examples can be found in Mardia and Jupp (1999). For example, when studying variables that influence the climate at a certain site, it is found that, from a meteorology point of view, most studies focus on wind direction and related variable such as rainfall (Carnicero, et al. 2011). Circular regression means any regression involving circular variables as response or predictor variables. In this paper, we investigate two circular regressions. One is called a linear-circular regression, which has a linear variable as response and a circular variable as predictor. The other is called a 40 ISSN 2424-6271 IASSL circular-linear regression, which has a circular response variable and a linear predictor variable.
In many applications, it is reasonable to assume that the regression function varies monotonically with the predictor variable in some region of interest. Order-restricted inference in a simple linear regression (Mukerjee and Tu, 1995) is only useful when the association between a response and predictor variables is weak in general. When the predictor or response variable is a circular variable, it is shown in this paper that the order restricted inference is useful, since a simple linear regression involving a circular variable and a linear variable tends to have a weak association. We show this phenomenon with a couple of examples in Section 4. The restricted inference is conservative and may produce a wider confidence interval than the unrestricted method, particularly when the magnitude of slope is large.
This paper is organized as follows. In Section 2, we explain the backgrounds that are useful for later sections. In Section 3, we propose order restricted circular regression models. In Section 4, we present data analysis examples, followed by the discussion and concluding remarks in Section 5.

Circular Regression
A variable that is measured in the form of any periodic manner is called a circular variable. For a couple of examples, an angle is a circular variable having 2π period, and the time of a day is a circular variable having 24 hours as the period. In this section, we discuss circular regression models that take into account of the periodic nature of circular variable. Suppose one observes ) , y where we assume that  is normally distributed with mean 0 and variance 2  and 2  is small enough so that 3 times the standard deviation is less than 1.

Restricted Inference in Simple Linear Regression
In this section, we present the results from (Mukerjee and Tu, 1995) about the order restricted simple linear regression. Consider a simple linear regression given by the following: Then, an U denote the lower and upper bounds, respectively, Considering inferences about the regression function at a given point 0

Circular Response Variable and Linear Predictor Variable
In this section, we study the case of regression having a circular variable  on a linear predictor variable Y . For example,  represents the spawning time of a particular fish and Y represents the tidal amplitude of the fish"s environment. A particular numerical example is given later in the example section. We propose the following regression frame work: where  represents a normally distributed error having zero mean and the small variance enough to cover the range of (-1, 1) using plus and minus of three times the standard deviation. In (2) Using those formulas in Section 2, we can obtain an % 100 ) 1 (   confidence interval for  as shown below, which is also displayed in Figure   1.

Linear Response Variable and Circular Predictor Variable
In this section, we consider the case of regression having a linear variable Y on a circular predictor variable  . For example,  may represent the month of year and Y represents a seasonal linear variable. A particular numerical example is given later in the example section. In this paper, we study a type of regression frame work shown in (1).

Circular-Linear Model
In a marine biology study by Robert T. Warner at University of California, Santa Barbara, data were gathered on the spawning time of a particular fish (Lund, 1999). It is hypothesized that the spawning time is affected by tidal characteristics of the fish's environment; For example as the amplitude goes down, the spawning time moves away from the mean direction time, which estimate is given by 13.42. In the following, we estimate the model (2)   Amplitude (Right) Figure 3: Confidence band with the mean line, where the 'cosine' and 'y' represent cosine of (spawning time -mean direction) and the amplitude of tides, respectively.
In Figure 3, we provide the confidence band with the mean line using the method in Section 2. In Table 1, those mean added values used in the plot are listed. Using (3), the corresponding "spawning times" for the mean added amplitude of tide equal to -20 is given by ISSN  In this section, we have shown that the restricted confidence interval is narrower than the unrestricted one.

Linear-Circular Model
Air quality is defined as a measure of the condition of air relative to the requirements of one or more biotic species, and/or to any human need or purpose at a given location. As the air quality index (AQI) increases, an increasingly large percentage of the population is likely to experience increasingly severe adverse health effects. AQI varies by pollutant, and is different in various geographic locations. De Wiest and Della Fiorentina (1975)  In Figure 5, we provide the mean line for those values in the sampled range of wind direction, along with the confidence band. Those mean added values used in the plot are listed in Table 2. represent the air quality index and cosine of (wind direction -mean direction), respectively. ISSN 2424-6271 IASSL Using (1), a 95% confidence interval of the corresponding "air quality index" for the wind direction equal to 75 degrees is given by (0.4581, 0.6834), whereas the unrestricted confidence interval is (0.4623, 0.6834). In this section, it is shown that the restricted confidence interval can be wider than the unrestricted one when the magnitude of slope is large.

Discussion and Concluding Remark
In this paper, we present the order-restricted inference on two types of circular regressions, where we utilize one example from ecological study and the other example from environmental study. In both examples, plots for mean values with confidence band are provided. Lacking pivotals, deriving confidence intervals in the restricted case is difficult. This constitutes perhaps the major deficiency of restricted inference at present as far as applications of the restricted methodology to applied problems are concerned. A systematic method consists of inverting one sided test of hypotheses (Lee 1984;Schoenfeld 1986;Williams 1977), and we used formulas derived using that method. As noted by Williams (1997) and Mukerjee and Tu (1995), these intervals are conservative, because least favourable distribution are used to compute the level of significance for composite null hypotheses in each of the one-sided test, and these distributions are usually different.

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We emphasize that the restricted inference is necessary in a situation where one observes a weak association and has knowledge of subject matter about the sign of slope. On the other hand, since the definition of "weak association" is somewhat indefinite, without clear knowledge about the sigh of slope, we recommend our readers to construct both restricted and unrestricted confidence intervals, and choose a better one to make an inference. Nonetheless, if one is clear about the sign, she/he needs to apply the restricted inference. We conclude this manuscript hoping that the models proposed in this paper benefit broad applications in diverse disciplines.