The Dynamics of Spatial Behavior: How can robust smoothing techniques help?

Itay Hen1, Anat Sakov2, Neri Kafkafi3, Ilan Golani4, Yoav Benjamini2

1. Department f Physics, Tel Aviv University, Israel

2. Department of Statistics and OR, Tel Aviv University, Israel 

3. Behavioral Neuroscience Branch, National Institute on Drug Abuse/IRP, Baltimore,   Maryland 

4. Department of Zoology, Tel Aviv University, Israel

Abstract

A variety of experimental setups and paradigms are currently employed in the neurosciences for automatically tracking the location of an animal in an experimental setup and for extracting features of interest out of it. Many of these features, however, especially those relating to the dynamics of movement, are critically sensitive to the unavoidable noise and artifacts of tracking. In this paper we examine the relevant properties of several common smoothing methods and suggest a combination of methods for retrieving locations and velocities and recognizing arrests from time series of coordinates of an animal’s center of gravity. We accomplish these by using robust nonparametric methods such as Running Median and locally weighted regression methods. The smoothed data may, subsequently, be segmented to obtain discrete behavioral units with proven ethological relevance. Significant new parameters such as the length, duration, maximal speed, and acceleration of these units provide a wealth of measures for, e.g., mouse behavioral phenotyping, studies on spatial orientation in vertebrates and invertebrates, and studies on rodent hippocampal function. This methodology may have implications for many tests and measurements of spatial behavior. It is implemented in a stand-alone publicly available program.

1. Introduction

In the Neurosciences, data on locomotor behavior, spatial orientation, navigation, spatial memory, and even social behavior often consist of a time series of coordinates representing the organism's location. Experimental Setups collecting such data include the Open Field Test, the Photobeam Cage, the Morris Swim Task, the Elevated Plus Maze, the Holeboard, and a variety of other spatial mazes. As elaborated in this study, the data acquired by these devices suffer from noise and artifact problems, which are inherent to all tracking systems and which critically affect the results. Noisy data deform the actual path traced by the animal, and have an even more devastating effect on the derivation of velocities and accelerations.
	 Filtering the raw data of the noise allows both a computation of velocities when the animal is moving, and an isolation of arrests (zero velocity) when it stays in place.  The smoothing thus uncovers an underlying "skeleton" consisting of an alternation of arrests and motion segments whose dynamic features can subsequently be analyzed. 
	Most of the studies performed in the above listed setups focus on the animal's location, ignoring velocity and acceleration (see however Tchernichovski and Golani, 1995; Tchernichovski et al., 1998; Pierce-Shimomura et al., 1999; Kafkafi et al., 2001; Whishaw et al., 2001; Wallace et al., 2002). 
	The benefits of an undistorted moment-to-moment record of velocity and acceleration cannot be overestimated: within a dynamic framework, the acceleration and velocity of the animal are the outcome of all the concurrent endogenous and exogenous “forces” acting upon it. Conversely, the attraction or repulsion exerted by a wall, a cliff, an edge, a familiar place, a partner, or a chemical gradient is revealed by the momentary values of these parameters. In rodent open field studies, for example, the forces exerted by the animal’s home base (Eilam and Golani, 1989), or any other familiar place (Tchernichovski et al., 1996), are reflected in the animal’s velocity and acceleration. The momentary velocity of an animal can tell us whether it “thinks” it is running away or toward its home base, or how familiar the immediate environment is (Tchernichovski and Golani, 1995; Tchernichovski et al., 1998), or (on the basis of the previous findings) what method of navigation it uses (Whishaw et al., 2001; Wallace et al., 2002). An automated high throughput acquisition of such data becomes possible, however, only after the data have been properly smoothed.
One ethologically-relevant point regarding velocities is that involving zero or close-to-zero velocities, i.e. stops.  An organism's locomotor behavior often consists of an alternation between progression and stopping, be it a nematode (Pierce-Shimomura et al., 1999), an insect (Miller, 1979; Collins et al., 1994), a fish (O’Brien et al., 1989), a lizard (Pietruska, 1986), a bird (Pienkowski, 1983), or a mammal (Kenagy, 1974; Golani et al., 1993). The movements it performs during a stop, be it foraging movements, scanning or movements related to any other form of information gathering, are reflected indirectly in the spatiotemporal properties of the stop (e.g.,Drai et al., 2000). Mouse inbred strains, for example, may differ substantially in the rate, type, rhythm, and number of scans they perform during a stop. These differences in the manner and intensity of information gathering are indirectly reflected in the duration, spatial spread, and average velocity of movement during stopping behavior (Drai et al., 2001). This is why characterizing the stop-and-go behavior should be both ethologically meaningful and results-wise fruitful. As we show here, the sometimes-erratic nature of animal movement requires appropriate smoothing methods. Furthermore, the smoothing methods appropriate when the animal is on the go become inappropriate when it stops. Therefore, a combination of methods must be used.
Finally, even a straightforward measure such as the overall distance traveled by the animal illustrates the difficulties inherent in the process of measurement. This is the primary measure used in most locomotor tests in mouse behavioral phenotyping studies and in psychopharmacological studies and pre-clinical screening of drugs (e.g., Cerbone and Sadile, 1994; Demarest et al., 1999; Strohle et al., 1998). This measure, however, is sensitive to the outliers typical to most tracking systems. For example, if a system tracks an animal at a rate of 10/s for a 1h session, and recognizes the animal by mistake at the other end of a 40 cm box in only 36 cases (an error rate of just 0.1%), the distance added by these outliers would be almost 30 meters, even if the animal did not move at all. This problem is dangerously ignored since many commercial tracking systems are “black boxes” that supply the final measures, but do not make it easy for the user to view the path coordinates and manipulate the algorithm.  
	While our approach is implemented in the present study on data borrowed from rodent locomotor and exploratory behavior, it clearly pertains to any study of an organism's unrestrained behavior: access to high throughput quantities of undistorted velocities and accelerations has wide implications for ethology and the various branches of behavioral neuroscience.
	In what follows we first describe the data acquisition process (section 2); then the different types of tracking errors (section 3) and the basic features of smoothing methods used in Statistics (section 4). Two of these methods, LOWESS and Repeated Running Medians, we use in a combined way to obtain robust smoothing without eliminating stops (sections 5,6). We end with the effects of appropriate smoothing (section 7), and with examples of usage of velocities in Behavioral Neuroscience (section 8).

2. Acquisition of rodent coordinate time series and the concept of smoothing 

In the open field test, a rodent is typically placed in an unfamiliar circular arena devoid of objects (Hall, 1934). With present day technology, as the rodent ambulates in the arena, a tracking system is following it while recording its location. In our work we use the EthoVision® system (Spink et al., 2001; Noldus et al., 2001) to track the animal at a rate of 25 (Pal Video System) or 30 (NTSC video system) records per second. The output of a 30 minutes session of such recordings is a file containing, respectively, 45,000 or 60,000 records of location - each involving the values at two coordinates.
The tracking system divides the arena into tiles or pixels. The more tiles there are the better the resolution is. Each record is the (x,y) location of a tile. Typically, the tracking system can only give one pixel at a given time, and since a pixel is usually much smaller than a rodent (pixel size is about 0.5-2 ), the system actually records the location of the rodent’s center of gravity.   
When rodents’ space coordinates time series is acquired by a tracking system, it is both natural and inevitable that the measured location is differnt than the actual location. Noise (i.e. any deviation from actual) would be added to the output data (for reasons to be elaborated in the next section). Whatever the level of the noise when measuring location is, when looking at velocity, which is the first derivative of location, the noise is even larger. Estimating velocity by differencing the locations measurements at succesive time points is noisier than the noise level in the measured location (Figure 1). The main reason for this distortion comes from probability theory, which dictates that the variance of the noise level of a difference is the sum of the variances of the individual noise.

Figure 1. Horizontal coordinates of location time series segment (left) and highly noisy velocities obtained by numerical differentiation of the time Series (right).

It is common practice to reduce the noise by smoothing. Instead of using the raw measured location data as estimates for the actual location at each time point, in smoothing, for each time point, all measured locations over a small time interval containing it are used. This can be done, for example, by taking their average or median. In this case, the value of an individual point that is higher than its neighbors is replaced by a smaller value; the value of an individual point that is smaller than its neighbors is replaced by a higher value. This naturally leads to a smoother location series, which is also reflected in the visual inspection of the path 
		Every smoothing technique also has its drawbacks. Most notably it may distort the actual path by producing too smooth a picture. If the true underlying process is actually smooth, as the trajectory of a moving mouse is, the path and velocity will not be much distorted while the noise will be reduced; but an actual rough path will become smooth. To overcome this problem, the level of smoothing can be controled in most methods, and therefore its choice becomes an important decision reflecting the goals of the analysis. A special complication in our data is that the smoothness of the process is not homogeneous along the period tracked, for example changing (sometimes abruptly) between stops and movements. 
	Having the above in mind, we must take into consideration the kinds of noise the system produces, as well as the impact of the smoothing technique applied to eliminate that noise, on the information we wish to retrieve 
	The two smoothing techniques we use are not new. What is unique in our work is the combination of two inherently different methods. Typically, a researcher applies one smoothing method, tuning it to suit best the data and the questions of interest.  However, because of the inhomogeneity of a rodent's locomotor behavior, manifested in progression segments and stops, a single smoothing technique is not appropriate for both tasks of evaluating velocities and stops.  

3. Difficulties in measuring velocities and identifying arrests: Noise

	All tracking systems make measurement errors. These errors, commonly referred to as “noise”, might severely impair the utility of behavioral endpoints, and thus decrease the ability to get discriminative and replicable results. This applies even to the simplest endpoints such as total distance traveled (Kafkafi et al., a, submitted). The sources of this noise can be divided into three main groups:
1.	 Precision level noise.
2.	Tracking system erratic noise, mostly in the form of outliers.
3.	Body wobble tracking.

	Identifying and reducing the impact of these three sources is not easy, but may be critical for the quality of the results. In the following subsections we describe and illustrate the three sources of noise. 

3.1 Precision level noise

	An important feature of every tracking system is its resolution. When looking at a field through a tracking system eyes, we actually see a paved area where each tile is a pixel recorded by the camera. This pixel is actually the resolution of the system, and since the tracking system cannot tell us where on a tile the mouse really stands, the data it records is quantized. In addition, since typically a pixel is smaller than a mouse (0.5-2 ), and since in this case the system still provides one pixel per record, this pixel summarizes the location of the entire body. The summary may be the calculated center of gravity of the animal, the center of a marker attached to the animal, or the center of a tracked part of the animal’s body. 
	Precision noise might give rise to two major problems.  If the center of the tracked object is not exactly on one pixel than even if the animal is standing, the system might load a different pixel each time (the tracking system vacillates between 2 neighboring pixels).
	To appreciate this problem all one has to do is place a toy mouse in the arena and track it. If the toy’s center of gravity is located between two pixels, the location data would vacillate irregularly between the two pixels as is demonstrated by the recording in Figure 2. 

Figure 2. Horizontal coordinates of a toy-mouse as recorded by the tracking system

	If a one-pixel distance on the screen involves a 1cm distance in the arena, the stationary toy-mouse might accumulate a considerable total distance traveled. When tracked at the necessary rate (for short stops to be captured) of 25 or 30 frames per second, such distance per frame would imply that the toy-mouse in Figure 2 has been traveling at a speed of 25-30cm/s (covering a distance of up to 15-18m per minute)!
	Another difficulty that might arise due to system precision is when the animal moves slowly. The quantization effect causes ‘slow movement’ segments to be interpreted by the tracking system as a series of standstill periods in adjacent pixels. This is due to the fact that the system loads the same location, disregarding where on the pixel the animal's or the tracked object’s center of gravity is. Looking at the sequence of locations of such ‘slow moving’ segments we might be misled to conclude that the animal alternated between stopping and progression although no such alternation occurred. In Figure 3, one can clearly see the ‘step like’ behavior of the recorded data, taken from a C57 mouse session. Examination of the corresponding video record reveals that the mouse has been moving constantly throughout the entire interval.

Figure 3. The x location of a slow moving c57
		As these examples illustrate, even simple measures, like total distance traveled and speed are vulnerable to tracking-system noise arising from the granulated pixel data, because they depend on more than one coordinate, so that the noise terms in the measuring of each coordinate add up. In general, as “endpoints” become more complex and depend on more coordinates, they will be more affected by noise, and it is very easy to reach a situation where the signal-to-noise ratio is lower than 1, even when the original noise appears to be small. Instead of solving this problem for each measure separately, it is clearly preferable to solve it once and for all on the measured data, and then apply any desired measure to the processed data.
3.2 Tracking system erratic noise - System outliers

		System outliers constitute a different, and sometimes even a more crucial problem. Outliers are observations ‘far’ from other neighboring (in time) observations. In our case, the source of this phenomenon is most likely due to some erratic behavior of the tracking system. 
	 	Intuitively speaking, the difference between noise and outliers is that outliers are something you would have immediately corrected with no scruples whatsoever, if you just happened to notice them. When the data are automatically tracked and analyzed, however, you might not be able to look for them and notice many of them. 
		An example of such an outlier is demonstrated in Figure 4, taken from a short tracking segment of a DBA mouse. 

Figure 4. X location time series of a DBA mouse’s introducing a single outlier

	Outliers might have, of course, a devastating effect on many endpoints like total distance traveled, mean spatial spread of segments, excursion length, progression segment length, etc. Such outliers also have undesired effects on velocities. The considerable difference between the outlier and the adjacent points may yield high velocities if the outlier is not ignored. 
		Moreover, outliers have a devastating effect on most smoothing methods that are used to deal with the previous problem of precision noise but not with outliers. Figure 5 shows the velocity estimations of the location time series presented in Figure 4, derived in two ways: by simple numerical differentiation and by a method called Local Polynomials smoothing (to be explained in Section 4).  Note that although the local polynomial smoother gives a more realistic picture of the mouse velocity, the effect of the outlier is nevertheless large.

Figure 5. Velocity vs. time plot of the path presented in Figure 4. Velocities were obtained by numerical differentiation (black) and by Local Polynomial smoothing (red)

3.3 Body wobble tracking

When interest is focused on obtaining the path or momentary velocity of a tracked animal, one must also take into consideration a third source of noise, which is due to the animal's own movements; we could call it body-wobble. By body wobble we refer to all movements of the animal that are not a part of its whole-body progression, for example head scanning or incipient shifts of weight. Body wobble consists of real movements that the animal performed, and therefore may, depending on the goal of the study, be the very subject of investigation.  However, for the purpose of obtaining the animal’s path, or its velocities, these movements are undesired side effects that must be separated and removed from the ‘progression movements’ of the animal, and thus should be treated as noise.
Typically, with a good tracking system, the size of an animal is larger than a pixel size. It is thus seen by the tracking system not as a single pixel, but as a group of pixels. In order to provide us with a single location of the animal, the tracking system performs some kind of averaging over these pixels, usually a computation of the pixels’ geometrical center. Movements like head scanning or shifts of body weight, even when the animal does not progress at all, might affect this group of pixels and hence its center.  The result of the body wobbling is that even when an animal is moving straight forward, the recorded track may exhibit deviations quite perpendicular to the main direction of progression. Thus, the over-sensitivity of the tracking system to small movements of the body becomes a drawback for the analysis of path and velocities on a more global scale. The misleading effects of body wobble can be seen in the path presented in Figure 6. In this figure, a DBA mouse is tracked arriving from the bottom and progressing in a counter-clockwise direction. The solid line is the recorded data. The effect of the DBA’s body wobble is clearly seen in the figure as jittery saw-tooth-like oscillations perpendicular to the direction of progression corresponding to the shifting of its weight as it is walking. 

Figure 6.  Raw (solid) and smoothed (dashed) path (xy plot) of a DBA mouse proceeding in a counterclockwise direction.

In order to eliminate these effects, once again, smoothing measures need to be taken. The dashed line shown in Figure 6 is the smoothed path of the DBA mouse, revealing its underlying true path. It is important to note that for the settings of the system used by us, body wobble noise is very similar to precision noise in terms of its magnitude, and thus, as it turns out, the smoothing techniques used to eliminate precision noise are appropriate also for the purpose of the elimination of body wobble noise. Let us comment, however, that this need not always be the case. For some settings, the effects of body wobble noise shadows precision noise or vice versa.  Upon tracking a turtle in our setup, for example, body wobble was larger than system noise, and in order to remove it, stronger smoothing had to be applied. 
It is important to remember, however, that for some purposes, body wobble may be an important or even the most important feature to be measured and investigated. For example, in the context of computing the properties of staying in place behavior (Drai et al., 2000; Drai et al., submitted; Kafkafi et al., b, submitted), one would be interested in sparing as many scanning traces as possible because they contribute to several measures of the animal's mobility during these episodes. In particular, mouse strains were found to differ significantly in the spatial spread and average velocity of their staying in place behavior because we were successful in not sifting out the larger amplitude body wobble movements (Drai et al., submitted; Kafkafi et al., a, submitted).

4. Smoothing techniques

		While it is a common statistical practice to reduce the noise by smoothing, not all smoothing methods were created equal.The current section gives a general overview of some of the most commonly used smoothing techniques. It illustrates the importance of using the proper technique in view of the goals, which are – retrieving path velocities and stops. 
		The most common smoothing technique used to reduce system noise is the Moving Average. This method uses a sliding window, shifting along the time series of coordinates from one data point to the next. The value of each data point is replaced by the average of the n points surrounding it, with (n-1)/2 points on each side. n is called the width of the smoothing "window" (see Box and Jenkins, 1970; Velleman and Hoaglin, 1981). The wider the window is, the more data points are used in the evaluation of each data point, leading to ‘stronger’ smoothing. Figure 7 provides a good example of Moving Average smoothing and the effect of different choices of window sizes. 

Figure 7. Horizontal location vs. Time plot of DBA mouse (black) smoothed by Moving Average with window sizes of  5 points (red) and of 20 points (blue)

	The average used in this method can be the simple average, giving equal weights to each data point in the window (as is done in Figure 7). A weighted average, which assigns large weight to data points in the center of the window, and gradually decreasing weights while getting further from the center, may alternatively be used. The result of such an operation is a smoothed time series, implying that noise is substantially reduced and, moreover, numerical differentiation of the smoothed series usually leads to more realistic velocities when compared to numerical differentiation of the un-smoothed (measured) locations. For the same window width the weighted average follows the original data more closely.
		From our point of view, however, there is a disadvantage to both. When applying them to retrieve velocities from the raw data, they lower speed peaks and increase the speed during slowing down or stopping.  Both are natural consequenses of smoothing, but while lowering peaks has little consequences in our case, increasing speed during slowing down has grieve consequences, as it turns the abrupt stops into slowing down episodes, sometimes to the point of entirely eliminating them.
	The problem is demonstrated in the following example. Consider the following coordinates of a mouse’s location, beginning with a 10 frames movement followed by a 4 frames stop, and ending with a 7 frames movement again. The first column of Table 1, shows the animal’s location. The 2nd column indicates whether, at each frame, the mouse moves or stops. Once applying a Moving Average smoother with a window size of 5 frames (2 frames on each side of the focal data point), one notices the undesired effect of the local averaging on the stop, which has disappeared!  Stops do not always vanish as in this example, but since the local averaging “lifts” the first few and last few locations of each stop, using local averaging generally eliminates very short stops and shortens the duration of longer stops. To summarize, using Moving Average is inappropriate for identifying stops. We will discuss another disadvantge of the Moving Average in a short while.

Location	True Move
/Stop	Smoothed (Averaged) Location	Conclusion: M/S	Frame Number	
36	Move	-	-	1	
31	Move	-	-	2	
27	Move	28.2	-	3	
24	Move	26.4	Move	4	
23	Move	23.8	Move	5	
27	Move	21.4	Move	6	
18	Move	19.2	Move	7	
15	Move	17	Move	8	
13	Move	13.6	Move	9	
12	Move	12	Move	10	
10	Stop	11	Move	11	
10	Stop	10.4	Move	12	
10	Stop	10.2	Move	13	
10	Stop	11	Move	14	
11	Move	12.2	Move	15	
14	Move	14	Move	16	
16	Move	16	Move	17	
19	Move	18	Move	18	
20	Move	-	-	19	
21	Move	-	-	20	

Table 1. Effects of Moving Average on identification of stops

Another commonly used smoothing technique is the Local Polynomials method (Fan and Gijbels, 1996). This method also uses a window of a pre-specified size around each data point in turn. However, instead of assigning each point the average of its surrounding neighbors (as the Moving Average does), this method uses a simple or weighted least squares fit of a low order polynomial (usually a line or a parabola) to the data points in each window. Once such a polynomial is found, the data point at the center of the window is replaced by the value of the polynomial at that point. 
This procedure enables us to determine the velocity at a point, not by numerical differentiation of the smoothed locations, but by using the derivative of the fitted polynomial at that point. 
	As in the Moving Average, the window size affects, visually, the smoothed sequence of locations. Furthermore, here the choice of the degree of the polynomial is also important: the higher the degree of the polynomial is, the weaker is the smoothing. The effect of the degree can best be seen in Figure 8 below. The gray line in the figure indicates the horizontal coordinates of a 3sec segment of a DBA mouse. The blue, red and black lines are the smoothed data obtained by using Local Polynomials smoothing with polynomials of the 1st (line), 2nd and a 6th degree, respectively. 

Figure 8. Horizontal location vs. Time plot of DBA mouse (black) smoothed by Local Polynomials of one (blue), two (red) and six (green) degrees (window size of 15 frames)

As can be seen in Figure 8, the degree of the polynomial chosen for the smoothing has a tremendous effect on the outcome. With a proper choice of parameters for the smoothing, this method is more flexible than the Moving Average, in the sense that it can accommodate better to the pattern of the data within a window. Although it somewhat reduces the problem of eliminating stops or shortening them, it would not solve the problem completely. 
Moreover, there is a second, not less crucial disadvantage that both methods share, namely, both are not robust, in the sense that they cannot deal well with system outliers. When fitting a Local Polynomials smoother to ‘outliers’ data, the outliers have a large impact on the resultant smoothed curve in the form of a ‘dent’ in the direction of the outlier. Figure 9a shows the impact of such a single outlier on the smoothed curves of Local Polynomials and Moving Average and the further implications of that impact on the velocities derived from them (Figure 9b). 

Figure 9a (left). Smoothed locations using Moving Average (red) and Local Polynomials (blue) smoothing of a time series with an outlier (black).
Figure 9b (right). Velocities derived from location smoothing: Moving Average (red) and Local Polynomials (blue).

The problem of outliers may be solved by using robust methods.

Still another set of smoothing tools is the smoothing splines (Hהrdle, 1991). When looking for a polynomial, which minimizes the residuals sum of squares (residual being the distance between the actual value at time t and its estimate using some smoothing method), then the higher the degree of the polynomial is, the smaller the residuals sum of squares becomes. However, the result is, typically, a high degree polynomial, which is not smooth. The idea of splines is to set some penalty, so that the higher degree polynomials get higher penalty. Smoothing splines balances between the two contradicting tasks of minimizing the residuals sum of squares when we consider a twice-differentiable function on the one hand, and minimizing the penalty on the other. The smoothing parameter in this case is the penalty (as opposed to window size before). Using a penalty which is equivalent in the amount of its smoothing to a window size in local polynomials generates similar looking sequences of smoothed locations. 
	Finally, the Fourier Transform approach to smoothing, known as the FFT after the algorithm carrying this name, takes a different direction (Efromovich, 1999). The original function is represented as a weighted sum of periodic functions, then the terms involving small weights are dropped and the result is back transformed to the original scale, producing a smoother function. The FFT is excellent for smoothing periodic functions but also useful for other smoothing functions. It nevertheless fails to cope with inhomogeneous functions of differing smoothness levels at different parts of the time series, as is the case with the path of the mouse. It is also not robust enough to cope with outliers. 

5. LOWESS - A robust smoothing technique

	As we have shown above, both Moving Average and Local Polynomials smoothers can be highly influenced by outliers. This influence is manifested in the distortion of the values of the measured locations as well as velocities in the neighborhood of an outlier. 
It is therefore preferable to have a smoothing method that is more robust to outliers. The Locally Weighted Scatter plot Smoothing (LOWESS) proposed by Cleveland (1977) is an iterative procedure, combining the ideas of Local Polynomials smoothing with robustness to outliers. As before, the procedure moves along the series of time-points, and for each, considers all its neighbors, and summarizes them using a local polynomial to produce the single estimated location at that time point.
 At the initial step, the weighted version of local polynomial is being used to get a first smooth estimate of the actual location - a first fit. Next, the closeness of the measured data to their fit is evaluated by calculating their differences, which are called residuals. A large residual indicates a poor fit at the point and suggests that the corresponding measured location is far from its neighbors, and so is less relevant to the fit. At the extreme, a very large residual indicates that the point is an outlier, and should have no effect on the current fit at all. This is achieved by introducing an additional set of weights, where the weight of each observation is maximal if its residual is zero and decreases as the residual increases in size. At the extreme, beyond some internally estimated threshold, the weight is set to zero, indicating that the observation has been automatically classified as an outlier. 
The new set of weights, based only on the residuals, is now combined with the previous set of weights that was based only on the time distance, by taking their product.  In the 2nd step, a local polynomial is fitted again, using now the combined set of weights, to produce the second fit. The above two steps are further iterated, taking the residuals from the second fit, to produce a new set of weights, etc. until the change from one iteration to the next diminishes. Practically, we found that repeating the two steps described above twice suffices.
		LOWESS solves both the precision noise problem and the outliers problem. LOWESS is being applied on the x-direction and the y-direction, separately. The velocity at each time point, and each direction, is then the derivative of the local polynomial at each time. Say, and  are the estimated velocities in the x and y directions at time t, respectively, then the total estimated velocity at time t is . An illustrative example of the operation of LOWESS is delayed to Section 7.
		Although robust, the LOWESS technique, much like the Local Polynomials technique, does not solve the problem of estimating the abrupt changes at very low velocities, and consequently it eliminates very short stops and shortens others. It is thus inappropriate for the task of identifying stops. To this end it would be preferable to use a method that applies only very local smoothing but is still robust. Such a method is discussed in the following section. 

6. Recognizing stops with repeated Running Medians 

	In light of the short stops elimination problem, and the chopping of longer stops edges as discussed in Section 4  (and demonstrated in Table 1), one seeks a local resistant smoother, which would not be affected by extreme observations inside the smoothing window (for identification of stops, extreme is an arbitrarily positive velocity). The simplest form of such a smoother is the Running Median (Tukey, 1977). The Running Median procedure is similar to the Moving Average, but instead of calculating the mean of observations within a neighnorhood (window), one considers their median, which is a robust estimator for outliers. This seemingly small change has enormous effect on the performance, as the median is a robust non-linear function of the data. In Table 2, we apply to the same data as in Table 1, the Running Median with window size of 5 frames. Note that, this time, the 4 frames stop is being identified correctly. Using the smoothed locations (3rd column) from the Running Median rather than the Moving Average produces a rougher curve of locations, which means that the Running Median is not good to estimate locations and hence the velocities. 

ocation	True Move
/Stop	Smoothed location (Median)	Conclusion: M/S	Frame Number	
36	Move	-	-	1	
31	Move	-	-	2	
27	Move	27	-	3	
24	Move	27	Stop	4	
23	Move	24	Move	5	
27	Move	23	Move	6	
18	Move	18	Move	7	
15	Move	15	Move	8	
13	Move	13	Move	9	
12	Move	12	Move	10	
10	Stop	10	Stop	11	
10	Stop	10	Stop	12	
10	Stop	10	Stop	13	
10	Stop	10	Stop	14	
11	Move	11	Move	15	
14	Move	14	Move	16	
16	Move	16	Move	17	
19	Move	19	Move	18	
20	Move	-	-	19	
21	Move	-	-	20	

Table 2. Effect of Running Median smoothing on identification of stops. 

Generally speaking, the disadvantage of the Running Median is that if there is a sequence of values repeating for at least n/2 times (at least half of the n data points in the window), then the median would be flat with that value. Visually, the resulting estimator is not so smooth and thus is not a recommended choice for evaluating velocities in general.  However, this disadvantage is useful for the purpose of identifying stops. A better method, which uses the advantages of the Running Median in order to identify stops, is the repeated Running Median, RRM (Tukey, 1977), which uses repeated iterations (applications) of Running Median with possibly different window sizes in each iteration. The identification of stops using this method is done by first smoothing the measured data with a Running Median. The next steps are smoothing the resultant smoothed data with another Running Median and so forth. Such repetitions have an effect of giving more weight to points closer to the center of the window. A proper choice of parameters (the window widths of the Running Median) yields correct identification of stops. The choice of the window sizes in the sequence was done by a trial-and-error, when the outcome was compared to the actual video, but also followed Tukey’s guidelines (Tukey, 1977) 
Once obtaining a final smoothed set of data points, sequences of equal values are then sought and marked as arrests. An illustrative example for the use of RRM is discussed in Section 7. 
The procedure as described above is being applied on the x-direction and the y-direction separately. An arrest is then defined when there is an arrest in both the x-direction and the y-direction at the same time. For more details see the Appendix.

7. Illustrative examples 

	At this point we wish to analyze two short segments, cut from a recorded 30 minute session of a DBA mouse ambulating on a 2.5m diameter circular arena. Tracking was performed using a video camera installed on the ceiling, right above the center of the arena, at a 2.5m distance from the floor, feeding directly into a PC computer running EthoVision® video tracking system (Spink et al, 2001; Noldus et al., 2001). We used a tracking rate of 25 frames/second in order to enable high time resolution, and the spatial resolution was about 1.3cm per video pixel. Coordinate files were exported out of EthoVision® and analyzed using SEE (Drai and Golani, 2001). Analysis of these segments with the RRM and LOWESS smoothers illustrates, on the one hand, the misleading effects of using measured data and inappropriate smoothing techniques, and on the other hand the efficiency of the combination of smoothers we have used. 

7.1 Identifying stops 	

	We start off with a 6.5 sec (~170 frames) time series of a DBA mouse. For the sake of simplicity we consider only the horizontal coordinate of its location, and attempt to retrieve stops (the same process is also performed on the vertical coordinate). Figure 10a shows the raw location data from the first segment to be analyzed. It is not hard to identify from the graph (or from watching the video) that during this short segment the mouse stopped five times. Note the quantization effect of the tracking system in the form of small upward and downward bumps causing the acquired trajectory to look a bit jittery. 

Figure 10a (left).  measured location data from a short segment.
Figure 10b (right). Velocity vs. time obtained by using direct numerical differentiation of raw data.

	As shown in Figure 10b, retrieving velocities from the measured location time series presented in Figure 10a by numerical differentiation of the raw data yields erratic and unrealistic velocities.   Attempting to identify stops in the figure by searching for zero velocity segments would show evidence for the first, third, and fifth stops, stops being represented in figure 10a by horizontal plateaus. Note, however, that there is no evidence of the second stop, and the fourth stop is shortened to a length that is hardly discernible. 
	Using Local Polynomials (2nd degree, window size of 15 frames) and Moving Average (window size of 15 frames), smoothing produces the smoothed locations shown in Figure 11a. 

Figure 11a (left). Smoothed location by Moving Average (red) and Local Polynomials (blue). The black line is the measured data.
Figure 11b (right). Velocities obtained from smoothed location from Moving Average (red) and Local Polynomials (blue).
 
	Clearly, the application of these smoothing techniques on the noisy measured data reduces the noise substantially. As shown in Figure 11b, the velocities obtained from the smoothed sets of data present a more realistic picture regarding velocities. However, as can be seen in this figure, both methods cause stops to look shorter than what they really are (Note again that the second and forth stops have been shortened). 
	Applying the RRM (with window sizes 7,7,5 and 3) produces the picture shown in Figure 12. The RRM, is not being used for evaluation of velocities, but only for identification of stops. It yields an output series in the form of a binary variable: thick black lines along the time axis indicating stops. Note the correlation between the stops as seen from the location data and from the RRM output.

Figure 12. Location raw data (black line) vs. time and the binary RRM output (thick black markings along time axis).

7.2 Dealing with jittery tracking and outliers

	For the 2nd example we consider another 6-seconds (150 frames) segment of a DBA mouse. Our goal is to retrieve the momentary velocities.  Figure 13a shows the measured location provided by the tracking system. Note the very jittery trajectory as well as a single outlier at the end of the 3rd second.  

Figure 13a (left). Measured location data from a short segment.
Figure 13b (right). Velocity vs. time obtained by numerical differentiation of measured location.

	Obtaining the momentary velocities by direct numerical differentiation of the measured location yields, as before, highly distorted results, as can be seen in Figure 13b. Note, that not only are the velocities obtained noisy, but also the effect of the outlier on velocities (~ -200 cm/s velocity spike followed by a positive ~ 200 cm/s spike within a very short time interval) is detrimental. 
	Using non-robust methods such as Moving Average and Local Polynomials smoothing (both with same parameters as before) yields a smoother and more realistic sequence of locations (Figure 14a). Yet, in both techniques, the smoothed location is being “pulled” downward by the outlier, lowering the true velocity values in the neighborhood of the outlier, demonstrating that both do not cope well with outliers. The velocities calculations based on the smoothed location of the same time series are shown in Figure 14b. The influence of the outliers - undesired peaks of velocity in the neighborhood of the outlier - is not absent in the graph. 

Figure 14a (left). Raw location data (black), Moving Average location (red) and Local polynomials location (blue).
Figure 14b (right). Velocity vs. time obtained by Moving Average (red) and Local Polynomials  (blue)

	Finally, using robust LOWESS (2nd degree polynomial, window size of 15 frames and 2 iterations) solves the problem, as can be seen in Figure 15a. This happens since LOWESS identifies the outlier and gives it a zero weight. The velocities obtained from LOWESS are shown in Figure 15b. Note that there are no residues whatsoever of the outlier and the resultant velocity is as desired. 

Figure 15a (left). Measured location (solid) and LOWESS smoothed location (dashed).
Figure 15b (right). Velocity obtained using LOWESS smoothing.

8. Applications 

Having made the point that adequate smoothing is a prerequisite for any analysis of a tracking record, and furthermore, having found appropriate measures to do so, we can now proceed to the potential inherent in the smoothed output. This robust information-rich sequence can now be studied vis-א-vis the correspondingly rich representations of the genetic, neurophysiological, or electrophysiological processes that mediate it.

 8.1 Applications in Behavior Neorogenetics and Behavioral Neuroscience.

	Rapid advances in bioinformatics have created a demand for tests of mouse behavior having high discriminatory power across strains and preparations, high replicability across laboratories, and high throughput for mass screening (Wahlsten et al., 2001).  In response to this demand, behavior geneticists now employ a variety of measures including a richer array of behavioral tests (Lederhandler and Schulkin 2000), standardization of these tests (Crabbe et al., 1999), and establishment of a Mouse Phenome Database (Paigen and Epig, 2000).  A complementary measure, suggested by our group, is automated recognition and measurement of a large  number of ethologically-relevant patterns reflecting motor, motivational and cognitive functions, all derived from half hour sessions of open field behavior (Drai and Golani, 2001, Kafkafi et al., 2001, Drai et al., 2001). This automation process is based on an algorithmic recognition of patterns, and it must be preceded by smoothing the data. 
To appreciate the effect of the smoothing consider the following two graphs (Figures 16a-b), which represent the same path and velocity trajectory traced by a DBA mouse in the course of half a minute (a video clip of this behavior can be downloaded at http://www.tau.ac.il/~ilan99). The path and the velocity-trajectory on the left are based on the measured location. From this type of noisy data one can only get the distance traveled, the proportions of that distance traveled along and away from the wall, and the amount of winding of the path (all contaminated with noise). In contrast, the path and velocity-trajectory on the right were obtained by smoothing the measured location along the principles outlined in the present paper. As illustrated, this sequence consists of an alternation of progression and lingering segments (Drai et al., 2000). Both segment types are discrete behavioral units with proven ethological relevance for rodents. They can be characterized by simple measures such as their length, duration, maximal speed, acceleration and other measures derived from these (see Figure 17). Treating the path as a string of discrete building blocks rather than a continuous series of coordinates allows a more straightforward analysis of complex structures (Drai and Golani, 2001; Kafkafi et al., 2001; Kafkafi et al., a and b, submitted; Drai et al., submitted). We presently have some 32 carefully designed endpoints that characterize these building blocks. 

Figure 16a (left). Un-smoothed path and trajectory. The yellow-to-red lines represent location and direction of progression. Azure lines designate speed trajectories.
Figure 16b (right). Smoothed path and trajectory. The yellow-to-red lines represent location and direction.. Azure lines represent speed, and empty spaces between segments designate the location and spatial spread of lingering episodes.

Figure 17. a time series of speeds including two progression segments (S1 and S2), and a lingering episode (LE) in between. The ratio between the maximal speed of a progression segment (vertical arrow in S1) and the segment’s duration (horizontal arrow in S1) describes a measure akin to segment acceleration (peak speed divided by duration). The horizontal line in LE designates median lingering velocity (from Kafkafi et al., b, submitted).

	Going back to the segmentation process: after smoothing with both the Running Median and LOWESS, the velocity time series output consists of motion segments - velocity "bursts" - performed on the background of arrests (segments of zero speed; Figure 17). Segmentation can be accomplished by obtaining the peak speed reached in every velocity burst. These peak values consist of at least two statistically distinct populations: One population of low velocities (the so-called lingering episodes) and one or several of higher velocities (known as progression segments). The mixture of these two natural populations can be sorted out into two or more Gaussians. The leftmost Gaussian consisting of lingering episodes. This can further be accomplished by using the EM algorithm (for an easy exposition see Everitt, 1981; for examples of the use of this algorithm see  Drai et al, 2000; Kafkafi et al, 2001). 

8.2 Applications to behavioral phenotyping of mouse behavior

	Having at hand sequences of progression segments and lingering episodes such as those illustrated in Figures 16 and 17, we propose for behavioral phenotyping a framework that is reminiscent of that used successfully in bioinformatics. In this framework, the path of the animal in the arena is considered as a structured and information-rich sequence that can be stored and reanalyzed. The raw coordinates from the open field sessions of many animals in many experiments and laboratories can be stored in an online database such as that offered by the Mouse Phenome Database (Paigen and Eppig, 2000). This database can be used for comparison of endpoint results from different laboratories, and for the development and testing of new endpoints, using See Path Smoother, See Path Segmentor, SEE and/or any other analysis software. Increasing the size and scope of this database should directly increase the potential power of the produced endpoints. Another possible advantage of this approach is that many experimenters can contribute data to such a database, while many data analysis specialists can study it without the need to conduct their own experiments. This framework can also be applied to other tests, such as the Morris water maze and the elevated plus maze, if they are conducted with video tracking. The experimenters can immediately employ any progress in the design of new endpoints for designing new experiments, and reevaluating old ones. This approach is thus more suitable for tackling the inherent complexity of behavioral phenotypes (Kafkafi et al., a, submitted). 

8.3 Application in studies of navigation

	Since velocity betrays the forces acting on the animal, it can be used to uncover attractors, such as familiar places in the environment, or repellors, such as the same places later on, when the animal becomes disinterested in them and avoids them (Tchernichovski and Golani, 1995; Tchernichovski et al., 1998).  In using this framework, Whishaw and colleagues (2001; 2002) compared the exploratory behavior of control rats and rats with damage to the fimbria-fornix.  They have shown that the higher velocity of the homeward portion of trips performed from a rat's home base (Eilam and Golani, 1989) requires an intact hippocampal formation. In using the distinction between progression segments and lingering episodes they have shown that while control and fimbria-fornix rats had similar outward segments, the last progression segment on the way home was significantly faster (and straighter) in the controls, when compared with that of the lesioned rats. This result was independent of testing in light or dark conditions, suggesting that rats employ dead reckoning navigational strategies to conclude the homeward portion of exploratory movements. Using the methodology outlined in this study, it is now possible to extend the analysis to whole-session velocity trajectories as well as to other derived measures (Kafkafi et al., b, submitted). This is important because i) the resolution and information content of these trajectories would finally now match that used in electrophysiological studies of the hippocampus (e.g., Best et al., 2001), enabling a common framework for lesion and electrophysiological studies. ii) The record would reflect not only the attractive properties of the home base during the first exposure to a novel environment, but also predictable changes (such as from attraction to repulsion) embodying spatial memory and habituation during later exposures ( Tchernichovski et al., 1998). 

8.4 Application in studies of Electrophysiology of the hippocampus

The repeated performance of lingering episodes in the same neighborhood in the environment, and the performance of long-distance arrivals to that neighborhood define an operational place for the animal (Eilam and Golani, 1989; Golani et al.,1993; Tchernichovski et al., 1998;   Drai et al., 2000  ; Drai and Golani, 2001; Whishaw et al., 2001). Could the hippocampal place-cells and their corresponding place-fields (e.g., Best et al., 2001; Wood et al., 2001; Mittelstaedt 2000) define locations, rather than operational places in the environment? In other words, could the place-cells monitor the animal's location regardless of whether the animal lingered or stopped repeatedly in particular locations? More generally, is there and what is the nature of a correspondence between behaviorally defined places and elecrophysiologically defined locations?  

Appendix

We outline the algorithms of LOWESS smoothing for finding locations and velocities and the RRM, which is used for recognizing stops. Location is composed of two coordinates, and the two algorithms are being applied separately for each coordinate. We then explain how to combine the two sets of smoothed locations to obtain velocity and identify stops. 

A.1 The LOWESS Algorithm

	The LOWESS algorithm requires several parameters:
1.	d - Polynomial degree. The same degree is used in each window. 
2.	h - The half width of the moving time window. 
3.	t - The number of iterations for the robust smoothing. 

For a given choice of these parameters and a given data set, :
Step A:
i. For each  define a neighborhood of size h, as follows:

ii.  For each , define weights for all :

Step B:
For each i, fit a d-th degree polynomial using weighted least squares with the weights defined in step Aii.  The fitted value at each i is denoted by  and is the value of the polynomial at that i (Note that for each i, there is another regression that should be fitted).

Step C:
i. Define the residuals to be . Let s be the median of .
ii. Assign another set of weights,  as follows: 

(Large residuals result in small weights and small residuals result in large weights).

Step D:
Repeat steps B and C t times but with  replacing .

The entire procedure including the initial computation and the iterations is referred to as robust locally weighted regression or LOWESS.

On the Choice of Smoothing Parameters: 

	Number of iterations (t) - The LOWESS can be repeated until convergence, or by specifying a number of iterations. We are using two iterations (t=2), which for most cases is sufficient.
	Window width (h) - The wider the window is, the smoother the output will be. There are statistical procedures for automatically choosing the width of the window (e.g. cross-validation, Fan and Gijbels, 1996). In our data, however, we are using h = 10. This choice came from trial and error where the output data looked relatively smooth on the one hand, and on the other hand, it still kept the important features of the data. This was done by trying different window widths and comparing the output to the actual video. This choice of bandwidth is untypical in the sense that each window contains about 0.02% of the data (on each side), while default values in commercial software (e.g. Splus) are around 25-33%.
	Polynomial degree (d) - The higher the degree of the polynomial is, the smoother the output is. Since each window is very short in time (less than a second), it is reasonable to assume that a mouse behavior is pretty smooth, reflecting a low degree polynomial. We are using a 2nd degree polynomial, as this allows us to estimate the acceleration as well as the velocity. Note that although the degree of the polynomial affects the amount of smoothing, the window width has a larger effect.

A.2  Estimating Velocities.

	In Step B of LOWESS, estimation of the local polynomial means estimating the d+1 coefficients of the polynomial. The derivative of the polynomial at each time, gives an estimate of the velocity. If   and  are the estimates of the velocities in directions X and Y respectively, at time t, we estimate the velocity at time t by . This is done for all t’s to get the velocities at all times.

A.3 Repeated Running Medians (RRM)

	The RRM algorithm in its role of identifying stops requires several parameters:
1.  - Window widths of the Running Median iterations.
2. l – Minimal length of arrests (to be explained below). 
3. - ‘Closeness’ parameter (to be explained below). 

The RRM algorithm is based on repeated iterations (applications) of Running Median smoothing, which requires a single parameter k, the window width of the smoothing. The Running Median algorithm is as follows: Given a data set , the smoothed location at time t, , is the median of the k s closest in time to .

For a given set of windows widths, , the RRM smoothes the data using the Running Median with window width, then it runs the Running Median on the smoothed sequence using , and so on r times. 

The RRM result is a sequence of smoothed locations. When a rodent stops, the smoothed sequence at the relevant time has the same value. We need to decide how many successive repetitions of the same values would be considered as a stop. This is done by another parameter, l. The last parameter needed is the ‘Closeness’ parameter  (a small number), determining how far from each other, within a window, data points can be, in order to still count them as the same point. With these choices the algorithm is as follows:
Step A: Run the RRM with given parameters to get the smoothed sequence.
Step B: Within the resultant data set, mark as stops only sequences of length l that keep the condition  for each  within that sequence. 	
This procedure is done separately on X and on Y. We identify a stop when the mouse does not progress on both directions at the same time.
The values of smoothing parameters we are using are: 
Window widths () - The sequence of windows widths for the RRM follows Tukey’s recommendations. The algorithm was verified against the video to make sure we identify stops correctly. We use 4 repetitions with window widths 7,5, 3 and 3.  
Minimal length of arrests (l) – 5 frames
Closeness parameter () – 0.0001

Acknowledgement     
	This study is part of the project “Phenotyping mouse exploratory behavior” supported by NIH 1 R01 NS40234-03. We thank Noldus Information Technology (http://www.noldus.com) for the use of their EthoVision system in Tel-Aviv University. SEE Path Smoother and other SEE related programs can be downloaded at  http://www.tau.ac.il/~ilan99

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