### Introduction

### Methods

### Participants

### Apparatus and procedure

*X*-

_{i}*T*│ /

*n*) between an actual hit position and the perceived/designated position was transmitted by using a custom made linking system that was previously devised. The size of magnitude and contact point with the rod applied point on the rod were also transmitted by the linking system and calculated as unit per SI.

### Analysis

*: Mean value and variance are measures of central tendency and variability for continuous (discrete) variables defined on the real line. They may also be applied to variables under certain circumstances. In general, statistics should be applied to variables to determine the center and the variability of probability densities and samples. Among several available measures, this analysis focuses mainly on the deterministic system, which state is described by a discrete numerical state variable. This method explains whether there is a consistent difference between the data. If so, a scientific guess can be made as to which data set belongs to what significant effect in experimental conditions.*

**Relative probability densities***h*is the number of occurrences of a repeating event-usually called frequency during the period of time. In statistics, this occurrence of an event (

_{j}*j*) is the number (

*k*) of times the event took place in an experiment.

*f*) is reflected by how often something happens divided by all outcomes. This event refers to the absolute frequency (

_{j}*h*) normalized by the total number of events (

_{j}*N*). The relative frequency, also known as empirical (experimental) probability is represented as follows;

*P*is the ratio of the number of outcomes (

_{j}*: We are always faced with uncertainty, ambiguity, and variability. Although we now have unprecedented access to information, being able to accurately predict the future is still almost impossible. The Monte Carlo simulation (based on the Markov chain model) allows us to see all possible outcomes and assess the possible impacts of the outcomes that may occur in the future, using not only information from the current state, but also from the sequence of events that preceded it. To address this likelihood, the haptic accuracy state variable was continuously placed as a probability density and variability according to the discrete time series. The relative ratio of the differences between all the designated conditions and the ratio of each condition were calculated using the Markov chain model based joint probability as follows;*

**Evolution of the probability***P*is the probability that the system is in the state (

*k*) at the time step (

*n*) and in the state (

*j*) at a next time step. The assumption for probability could not express how to create probabilities for combined events such as

*P*[

*A*∩

*B*] or for the likelihood of an event

*A*, given that it is known that event

*B*occurs. For instance, let

*A*be the state of distribution effect accuracy (within

*SD*±1) at one condition and

*B*be the state of non-distribution effect accuracy for the other condition. Does knowledge of haptic accuracy at the one condition change the belief that it will still be accurate for the other condition? That is,

*P*[

*B*] , is the probability that the accuracy of the other condition is ignoring information on whether it is accurate at the condition. This differs from, the probability that it is accurate at the other condition given that it is accurate at the time step (called conditional probability of

*B*given

*A*). It is quite likely that

*P*[

*B*] and

*P*[

*B*|

*A*] are different.

*X*

_{1}) and (

*X*

_{2}) for a participant. Using this information, the time-evolution of a system can be modeled to occupy each of a countable number of states (discrete set of states) about a continuous time step, where switching between states is treated probabilistically. In this case, the master that describes the evolution of the probability

*p*(

*j*,

*t*), where t is time, is defined by

*w*(

*j*←

*k*) denotes the transition rate from

*k*to

*j*. This modeling will allow the progression of haptic accuracy dependency (progression of performance probability) as an application. Intuitively, what can we say about this evolutionary probability

*p*(

*j*,

*t*)] increase monotonically as a function of time, whether it increase or decrease depends on the other probabilities [i.e.,

*p*(1,

*t*),

*p*(2,

*t*)⋯

*p*(

*n*,

*t*)]. Second, the probability

*p*(

*j*,

*t*) increase monotonically as a function of time or does not change with time. That is,

*p*(

*j*,

*t*) does not decay as a function of time and this is irrepspective of the values of the other probabilities. Third, the probability

*p*(

*j*,

*t*) decrease monotonically as a function of time or does not change with time. That is, does not increase as a function of time and this is irrespective of the values of the other probabilities. This observation, then, wil represent that the dominant state between conditions which influence would be more robust.

### Results

### Relative probability

*η*

^{2}13.475 (

*p*< 0.001), (Pearson Correlation R = - .61)]. In addition, these results also reflect the deviation of the performance parameter for the interaction between error and variance of the error from the different distances [F(2, 71) = 28,865, =

*η*

^{2}14.433 (

*p*< 0.001), (Pearson Correlation R = - .63)] (see Appendix 4 for the statistical approaches). The analyses show that observers influenced not only the stimulus magnitudes distinguished by the coefficient of restitution, but also the distributions of the encoded impressions by the distance from the hand to the impact (see Table 1).

##### Table 1.

__Table 2.__

### Evolution of the probability

*η*

^{2}.549 (

*p*< 0.001). As it is, the time-evolution of a system was analyzed to occupy each one of a countable number of states (discrete set of states) about a continuous time step, and where switching between states is treated probabilistically.

##### Table 3.

^{} Note: Proportion (percentage) of occurrences of the outcome in the statistical ensemble is given using the vector matrix and component notation. Proving for the error variance (30 cm) cases; [P(x=H)=49% P(30cm|H)=49%=0.32=P(30cm |H) *P(H)=0.32*0.49=0.16], [P(x=D)=51% P(30cm|D)=51%= 0.15= P(30cm|D)*P(D)=0.15*0.51=0.07]. Vector and matrix notation:

##### Table 4.

^{} Note: Model for the analysis, we used Master Equation as given (Eq. 7). Calculated ‘rate in’=w(1←2)P(2,t)+w(1←3)P(3,t), and calculated ‘rate out’=w(2←1)P(1,t)+w(3←1)P(1,t).

### Discussion

*: Here, one of the basic properties can explicitly propose a progressed step in the form of simple physical systems as indicated below. Apparent perception can be described as a function of the variable and constant parameter (*

**Physical property (Inertia tensor)***ψ*=

*Kϕ*). Where

^{β}*ψ*is apparent perception,

*ϕ*is the variable,

*β*is the slope, and

*K*is a proportionally constant parameter. However, the perception can be determined by a function of the constant (not variable) if given the same variables (

*ψ*=

*b*-

*a*log

*V*) (Stevens & Rubin, 1970). Where

*ψ*is the perceived impression (i.e., heaviness).

*V*is the constant,

*a*is the coefficient slope, and

*b*is the additive constant.

*ψ*≈

*M*× (

*CM*-

*O*)

^{2}) (Amazeen & Turvey, 1996). Where

*ψ*is perceived heaviness,

*M*is the mass of the object,

*C*is the location of the object’s center of mass, and

_{M}*O*is the location of the rotation point. Under this assumption, the perceived impression is uniquely constrained by the eigenvalues of the inertia tensor (Amazeen & Turvey, 1996; Carellro, Thuot, Anderson & Turvey, 1999; Stroop, Turvey, Fitzpatrick & Carello, 2000; Streit, Shockley & Riley, 2007; Shockley Morris & Riley, 2007; Harrison, Hajnal, Goodman, Isenhower & Show, 2011).

*A*=

_{p}*C*× (

_{e}*CP*-

*O*)

^{2}]. Where

*A*is the perceived spatial accuracy,

_{p}*C*is coefficient restitution of the object,

_{e}*CP*is the location of the center of pressure, and

*O*is the location of the rotation (grasp) point.

*CP*) or, in other words, a logarithmic function uniquely constrained by the eigenvalues of the inertia tensor [

*A*≒

_{p}*I*,

*I*=

*mass*× (

*CM*-

*O*)

^{2}]. We now know that

*A*≒

_{p}*I*. This, in fact, is the formal definition of the property used by the S-W illusion investigation (Davis & Brickett1977; Charpentier, 1981), what is called a precise control over the grasped non-visible objects (Amazeen & Turvey, 1996). The present study found that the property takes on a particular patterning in a comprehensive extended spatial haptic. The conjecture is that for a given applied pressure at the same impulse magnitude (=

*C*) as the parameter variable, haptic spatial accuracy (=

_{e}*A*) will increase as the rotational inertia (=

_{p}*I*) increase.

*: The point of view that the study established here holds some possibility of a better understanding of the haptic system. The assumption is that haptic accuracy traces are associated not only with physical traces, but also with the system of psychological accuracy coordinates. By this, the present study does not mean that the accuracy has one particular subsystem in haptic perception. On the contrary, these experiments conclusively show that this is not the case. Rather, when the haptic accuracy trace is formed, it is integrated with invariant characteristics of the physical property called rotational inertia, which gives it position in relation to the other (pressure distribution) associated psychological traces (Kinsella-Shaw & Turvey, 1992).*

**Psychological Property (estimating two conflicted inputs)***S*is the physical property being estimated, and is the operation by which the nervous system does the estimation (refers to different cues within a modality). In a perceived condition that is conflicted between different modalities, haptic minimal variance in the final estimation determines the degree to which modality dominates because of the effect of bias between inputs, where minimal variance in the final estimation determines the degree to which modalities dominate (

*W*is the sensor estimates weighted by their normalized reciprocal variances,

*c*is the variance of the final cutaneous (stimulus magnitude as a coefficient restitution) estimate, and

*k*is the variance of the final kinesthetic (stimulus distribution as an inertia tensor) estimate.